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شائع علم المثلثات >

arcsin(6x)+arcsin(6sqrt(3x))=-pi/2

الحلّ

arcsin (6 x) + arcsin (6 3 x) = - \frac{π}{2}

الحلّ

x \in R لا يوجد حلّ لـ

خطوات الحلّ

arcsin (6 x) + arcsin (6 3 x) = - \frac{π}{2}

Rewrite using trig identities

arcsin (6 x) + arcsin (6 3 x)

arcsin (s) + arcsin (t) = arcsin (s 1 - t^{2} + t 1 - s^{2})

Eq u a t i o n 0 :

متطابقة تحويل الجمع لضرب

= arcsin (6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2})

arcsin (6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2}) = - \frac{π}{2}

Apply trig inverse properties

arcsin (6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2}) = - \frac{π}{2}

arcsin (x) = a \Rightarrow x = sin (a)

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} = sin (- \frac{π}{2})

sin (- \frac{π}{2}) = - 1

sin (- \frac{π}{2})

sin (- x) = - sin (x) :

استخدم القانون التالي

sin (- \frac{π}{2}) = - sin (\frac{π}{2})

= - sin (\frac{π}{2})

استخدم المتطابقة الأساسيّة التالية:

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

= - 1

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} = - 1

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} = - 1

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} = - 1

حلّ:

x \in R

لا يوجد حلّ لـ

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} = - 1

Remove square roots

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} = - 1

من الطرفين

6 3 x 1 - (6 x)^{2}

اطرح

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} - 6 3 x 1 - (6 x)^{2} = - 1 - 6 3 x 1 - (6 x)^{2}

بسّط

6 1 - (6 3 x)^{2} x = - 1 - 6 3 x 1 - (6 x)^{2}

ربّع الطرفين:

36 x^{2} - 3888 x^{3} = 1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3}

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} = - 1

(6 1 - (6 3 x)^{2} x)^{2} = (- 1 - 6 3 x 1 - (6 x)^{2})^{2}

(6 1 - (6 3 x)^{2} x)^{2}

وسّع:

36 x^{2} - 3888 x^{3}

(6 1 - (6 3 x)^{2} x)^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= 6^{2} x^{2} (1 - (6 3 x)^{2})^{2}

(1 - (6 3 x)^{2})^{2} : 1 - (6 3 x)^{2}

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

= ((1 - (6 3 x)^{2})^{\frac{1}{2}})^{2}

(a^{b})^{c} = a^{b c}

:فعّل قانون القوى

= (1 - (6 3 x)^{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{1 \cdot 2}{2}

2 :

إلغ العوامل المشتركة

= 1

= 1 - (6 3 x)^{2}

= 6^{2} (1 - (6 3 x)^{2}) x^{2}

6^{2} = 36

= 36 (1 - (6 3 x)^{2}) x^{2}

36 (1 - (6 3 x)^{2}) x^{2}

وسّع:

36 x^{2} - 3888 x^{3}

36 (1 - (6 3 x)^{2}) x^{2}

(6 3 x)^{2} = 6^{2} \cdot 3 x

(6 3 x)^{2}

3 x = 3 x

3 x

a \geq 0, b \geq 0

بافتراض أنّ

n ab = n a n b

:فعّل قانون الجذور

= 3 x

= (63 x)^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= 6^{2} (3)^{2} (x)^{2}

(3)^{2} : 3

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

= (3^{\frac{1}{2}})^{2}

(a^{b})^{c} = a^{b c}

:فعّل قانون القوى

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{1 \cdot 2}{2}

2 :

إلغ العوامل المشتركة

= 1

= 3

= 6^{2} \cdot 3 (x)^{2}

(x)^{2} : x

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

= (x^{\frac{1}{2}})^{2}

(a^{b})^{c} = a^{b c}

:فعّل قانون القوى

= x^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{1 \cdot 2}{2}

2 :

إلغ العوامل المشتركة

= 1

= x

= 6^{2} \cdot 3 x

= 36 x^{2} (- 6^{2} \cdot 3 x + 1)

6^{2} \cdot 3 x = 108 x

6^{2} \cdot 3 x

6^{2} = 36

= 36 \cdot 3 x

36 \cdot 3 = 108 :

اضرب الأعداد

= 108 x

= 36 x^{2} (- 108 x + 1)

= 36 x^{2} (1 - 108 x)

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = 36 x^{2}, b = 1, c = 108 x

= 36 x^{2} \cdot 1 - 36 x^{2} \cdot 108 x

= 36 \cdot 1 \cdot x^{2} - 36 \cdot 108 x^{2} x

36 \cdot 1 \cdot x^{2} - 36 \cdot 108 x^{2} x

بسّط:

36 x^{2} - 3888 x^{3}

36 \cdot 1 \cdot x^{2} - 36 \cdot 108 x^{2} x

36 \cdot 1 \cdot x^{2} = 36 x^{2}

36 \cdot 1 \cdot x^{2}

36 \cdot 1 = 36 :

اضرب الأعداد

= 36 x^{2}

36 \cdot 108 x^{2} x = 3888 x^{3}

36 \cdot 108 x^{2} x

36 \cdot 108 = 3888 :

اضرب الأعداد

= 3888 x^{2} x

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

x^{2} x = x^{2 + 1}

= 3888 x^{2 + 1}

2 + 1 = 3 :

اجمع الأعداد

= 3888 x^{3}

= 36 x^{2} - 3888 x^{3}

= 36 x^{2} - 3888 x^{3}

= 36 x^{2} - 3888 x^{3}

(- 1 - 6 3 x 1 - (6 x)^{2})^{2}

وسّع:

1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3}

(- 1 - 6 3 x 1 - (6 x)^{2})^{2}

(a - b)^{2} = a^{2} - 2 ab + b^{2}

:فعّل صيغة الضرب المختصر

a = - 1, b = 6 3 x 1 - (6 x)^{2}

= (- 1)^{2} - 2 (- 1) \cdot 6 3 x 1 - (6 x)^{2} + (6 3 x 1 - (6 x)^{2})^{2}

(- 1)^{2} - 2 (- 1) \cdot 6 3 x 1 - (6 x)^{2} + (6 3 x 1 - (6 x)^{2})^{2}

بسّط:

1 + 12 3 x 1 - (6 x)^{2} + 363 x 1 - (6 x)^{2}

(- 1)^{2} - 2 (- 1) \cdot 6 3 x 1 - (6 x)^{2} + (6 3 x 1 - (6 x)^{2})^{2}

- (- a) = a

فعّل القانون

= (- 1)^{2} + 2 \cdot 1 \cdot 6 3 x 1 - (6 x)^{2} + (6 3 x 1 - (6 x)^{2})^{2}

(- 1)^{2} = 1

(- 1)^{2}

زوجيّ n

إذا تحقّق أنّ

, (- a)^{n} = a^{n}

:فعّل قانون القوى

(- 1)^{2} = 1^{2}

= 1^{2}

1^{a} = 1

فعّل القانون

= 1

2 \cdot 1 \cdot 6 3 x 1 - (6 x)^{2} = 12 3 x 1 - (6 x)^{2}

2 \cdot 1 \cdot 6 3 x 1 - (6 x)^{2}

2 \cdot 1 \cdot 6 = 12 :

اضرب الأعداد

= 12 3 x 1 - (6 x)^{2}

(6 3 x 1 - (6 x)^{2})^{2} = 363 x 1 - (6 x)^{2}

(6 3 x 1 - (6 x)^{2})^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= 6^{2} (3 x)^{2} (1 - (6 x)^{2})^{2}

(3 x)^{2} : 3 x

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

= ((3 x)^{\frac{1}{2}})^{2}

(a^{b})^{c} = a^{b c}

:فعّل قانون القوى

= (3 x)^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{1 \cdot 2}{2}

2 :

إلغ العوامل المشتركة

= 1

= 3 x

= 6^{2} \cdot 3 x (1 - (6 x)^{2})^{2}

(1 - (6 x)^{2})^{2} : 1 - (6 x)^{2}

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

= ((1 - (6 x)^{2})^{\frac{1}{2}})^{2}

(a^{b})^{c} = a^{b c}

:فعّل قانون القوى

= (1 - (6 x)^{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{1 \cdot 2}{2}

2 :

إلغ العوامل المشتركة

= 1

= 1 - (6 x)^{2}

= 6^{2} \cdot 3 x (1 - (6 x)^{2})

6^{2} = 36

= 36 \cdot 3 x (1 - (6 x)^{2})

= 1 + 12 3 x 1 - (6 x)^{2} + 36 \cdot 3 x (1 - (6 x)^{2})

= 1 + 12 3 x 1 - (6 x)^{2} + 36 \cdot 3 x (1 - (6 x)^{2})

1 + 12 3 x 1 - (6 x)^{2} + 36 \cdot 3 x (1 - (6 x)^{2})

وسّع:

1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3}

1 + 12 3 x 1 - (6 x)^{2} + 36 \cdot 3 x (1 - (6 x)^{2})

12 3 x 1 - (6 x)^{2} = 123 x 1 - 36 x^{2}

12 3 x 1 - (6 x)^{2}

3 x = 3 x

3 x

a \geq 0, b \geq 0

بافتراض أنّ

n ab = n a n b

:فعّل قانون الجذور

= 3 x

= 123 x - (6 x)^{2} + 1

1 - (6 x)^{2} = 1 - 36 x^{2}

1 - (6 x)^{2}

(6 x)^{2} = 36 x^{2}

(6 x)^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= 6^{2} x^{2}

6^{2} = 36

= 36 x^{2}

= 1 - 36 x^{2}

= 123 x - 36 x^{2} + 1

36 \cdot 3 x (1 - (6 x)^{2}) = 108 x (1 - 36 x^{2})

36 \cdot 3 x (1 - (6 x)^{2})

(6 x)^{2} = 36 x^{2}

(6 x)^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= 6^{2} x^{2}

6^{2} = 36

= 36 x^{2}

= 36 \cdot 3 x (- 36 x^{2} + 1)

36 \cdot 3 = 108 :

اضرب الأعداد

= 108 x (- 36 x^{2} + 1)

= 1 + 123 x - 36 x^{2} + 1 + 108 x (- 36 x^{2} + 1)

108 x (1 - 36 x^{2})

وسٌع:

108 x - 3888 x^{3}

108 x (1 - 36 x^{2})

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = 108 x, b = 1, c = 36 x^{2}

= 108 x \cdot 1 - 108 x \cdot 36 x^{2}

= 108 \cdot 1 \cdot x - 108 \cdot 36 x^{2} x

108 \cdot 1 \cdot x - 108 \cdot 36 x^{2} x

بسّط:

108 x - 3888 x^{3}

108 \cdot 1 \cdot x - 108 \cdot 36 x^{2} x

108 \cdot 1 \cdot x = 108 x

108 \cdot 1 \cdot x

108 \cdot 1 = 108 :

اضرب الأعداد

= 108 x

108 \cdot 36 x^{2} x = 3888 x^{3}

108 \cdot 36 x^{2} x

108 \cdot 36 = 3888 :

اضرب الأعداد

= 3888 x^{2} x

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

x^{2} x = x^{2 + 1}

= 3888 x^{2 + 1}

2 + 1 = 3 :

اجمع الأعداد

= 3888 x^{3}

= 108 x - 3888 x^{3}

= 108 x - 3888 x^{3}

= 1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3}

= 1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3}

36 x^{2} - 3888 x^{3} = 1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3}

36 x^{2} - 3888 x^{3} = 1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3}

من الطرفين

108 x - 3888 x^{3}

اطرح

36 x^{2} - 3888 x^{3} - (108 x - 3888 x^{3}) = 1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3} - (108 x - 3888 x^{3})

بسّط

36 x^{2} - 108 x = 123 x 1 - 36 x^{2} + 1

من الطرفين

1

اطرح

36 x^{2} - 108 x - 1 = 123 x 1 - 36 x^{2} + 1 - 1

بسّط

36 x^{2} - 108 x - 1 = 123 x 1 - 36 x^{2}

ربّع الطرفين:

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x - 15552 x^{3}

36 x^{2} - 3888 x^{3} = 1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3}

(36 x^{2} - 108 x - 1)^{2} = (123 x 1 - 36 x^{2})^{2}

(36 x^{2} - 108 x - 1)^{2}

وسّع:

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1

(36 x^{2} - 108 x - 1)^{2}

(36 x^{2} - 108 x - 1)^{2} = (36 x^{2} - 108 x - 1) (36 x^{2} - 108 x - 1)

= (36 x^{2} - 108 x - 1) (36 x^{2} - 108 x - 1)

(36 x^{2} - 108 x - 1) (36 x^{2} - 108 x - 1)

وسٌع:

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1

(36 x^{2} - 108 x - 1) (36 x^{2} - 108 x - 1)

فعّل قانون ضرب الأقواس

= 36 x^{2} \cdot 36 x^{2} + 36 x^{2} (- 108 x) + 36 x^{2} (- 1) + (- 108 x) \cdot 36 x^{2} + (- 108 x) (- 108 x) + (- 108 x) (- 1) + (- 1) \cdot 36 x^{2} + (- 1) (- 108 x) + (- 1) (- 1)

فعّل قوانين سالب-موجب

+ (- a) = - a, (- a) (- b) = ab

= 36 \cdot 36 x^{2} x^{2} - 36 \cdot 108 x^{2} x - 36 \cdot 1 \cdot x^{2} - 108 \cdot 36 x^{2} x + 108 \cdot 108 xx + 108 \cdot 1 \cdot x - 1 \cdot 36 x^{2} + 1 \cdot 108 x + 1 \cdot 1

36 \cdot 36 x^{2} x^{2} - 36 \cdot 108 x^{2} x - 36 \cdot 1 \cdot x^{2} - 108 \cdot 36 x^{2} x + 108 \cdot 108 xx + 108 \cdot 1 \cdot x - 1 \cdot 36 x^{2} + 1 \cdot 108 x + 1 \cdot 1

بسّط:

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1

36 \cdot 36 x^{2} x^{2} - 36 \cdot 108 x^{2} x - 36 \cdot 1 \cdot x^{2} - 108 \cdot 36 x^{2} x + 108 \cdot 108 xx + 108 \cdot 1 \cdot x - 1 \cdot 36 x^{2} + 1 \cdot 108 x + 1 \cdot 1

- 36 \cdot 108 x^{2} x - 108 \cdot 36 x^{2} x = - 2 \cdot 108 \cdot 36 x^{2} x :

اجمع العناصر المتشابهة

= 36 \cdot 36 x^{2} x^{2} - 2 \cdot 108 \cdot 36 x^{2} x - 36 \cdot 1 \cdot x^{2} + 108 \cdot 108 xx + 108 \cdot 1 \cdot x - 1 \cdot 36 x^{2} + 1 \cdot 108 x + 1 \cdot 1

108 \cdot 1 \cdot x + 1 \cdot 108 x = 2 \cdot 1 \cdot 108 x :

اجمع العناصر المتشابهة

= 36 \cdot 36 x^{2} x^{2} - 2 \cdot 108 \cdot 36 x^{2} x - 36 \cdot 1 \cdot x^{2} + 108 \cdot 108 xx + 2 \cdot 1 \cdot 108 x - 1 \cdot 36 x^{2} + 1 \cdot 1

- 36 \cdot 1 \cdot x^{2} - 1 \cdot 36 x^{2} = - 2 \cdot 1 \cdot 36 x^{2} :

اجمع العناصر المتشابهة

= 36 \cdot 36 x^{2} x^{2} - 2 \cdot 108 \cdot 36 x^{2} x - 2 \cdot 1 \cdot 36 x^{2} + 108 \cdot 108 xx + 2 \cdot 1 \cdot 108 x + 1 \cdot 1

36 \cdot 36 x^{2} x^{2} = 1296 x^{4}

36 \cdot 36 x^{2} x^{2}

36 \cdot 36 = 1296 :

اضرب الأعداد

= 1296 x^{2} x^{2}

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

x^{2} x^{2} = x^{2 + 2}

= 1296 x^{2 + 2}

2 + 2 = 4 :

اجمع الأعداد

= 1296 x^{4}

2 \cdot 108 \cdot 36 x^{2} x = 7776 x^{3}

2 \cdot 108 \cdot 36 x^{2} x

2 \cdot 108 \cdot 36 = 7776 :

اضرب الأعداد

= 7776 x^{2} x

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

x^{2} x = x^{2 + 1}

= 7776 x^{2 + 1}

2 + 1 = 3 :

اجمع الأعداد

= 7776 x^{3}

2 \cdot 1 \cdot 36 x^{2} = 72 x^{2}

2 \cdot 1 \cdot 36 x^{2}

2 \cdot 1 \cdot 36 = 72 :

اضرب الأعداد

= 72 x^{2}

108 \cdot 108 xx = 11664 x^{2}

108 \cdot 108 xx

108 \cdot 108 = 11664 :

اضرب الأعداد

= 11664 xx

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

xx = x^{1 + 1}

= 11664 x^{1 + 1}

1 + 1 = 2 :

اجمع الأعداد

= 11664 x^{2}

2 \cdot 1 \cdot 108 x = 216 x

2 \cdot 1 \cdot 108 x

2 \cdot 1 \cdot 108 = 216 :

اضرب الأعداد

= 216 x

1 \cdot 1 = 1

1 \cdot 1

1 \cdot 1 = 1 :

اضرب الأعداد

= 1

= 1296 x^{4} - 7776 x^{3} - 72 x^{2} + 11664 x^{2} + 216 x + 1

- 72 x^{2} + 11664 x^{2} = 11592 x^{2} :

اجمع العناصر المتشابهة

= 1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1

= 1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1

= 1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1

(123 x 1 - 36 x^{2})^{2}

وسّع:

432 x - 15552 x^{3}

(123 x 1 - 36 x^{2})^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= 1 2^{2} (3)^{2} (x)^{2} (1 - 36 x^{2})^{2}

(3)^{2} : 3

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

= (3^{\frac{1}{2}})^{2}

(a^{b})^{c} = a^{b c}

:فعّل قانون القوى

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{1 \cdot 2}{2}

2 :

إلغ العوامل المشتركة

= 1

= 3

= 1 2^{2} \cdot 3 (x)^{2} (1 - 36 x^{2})^{2}

(x)^{2} : x

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

= (x^{\frac{1}{2}})^{2}

(a^{b})^{c} = a^{b c}

:فعّل قانون القوى

= x^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{1 \cdot 2}{2}

2 :

إلغ العوامل المشتركة

= 1

= x

= 1 2^{2} \cdot 3 x (1 - 36 x^{2})^{2}

(1 - 36 x^{2})^{2} : 1 - 36 x^{2}

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

= ((1 - 36 x^{2})^{\frac{1}{2}})^{2}

(a^{b})^{c} = a^{b c}

:فعّل قانون القوى

= (1 - 36 x^{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{1 \cdot 2}{2}

2 :

إلغ العوامل المشتركة

= 1

= 1 - 36 x^{2}

= 1 2^{2} \cdot 3 x (1 - 36 x^{2})

بسّط

= 432 x (1 - 36 x^{2})

432 x (1 - 36 x^{2})

وسّع:

432 x - 15552 x^{3}

432 x (1 - 36 x^{2})

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = 432 x, b = 1, c = 36 x^{2}

= 432 x \cdot 1 - 432 x \cdot 36 x^{2}

= 432 \cdot 1 \cdot x - 432 \cdot 36 x^{2} x

432 \cdot 1 \cdot x - 432 \cdot 36 x^{2} x

بسّط:

432 x - 15552 x^{3}

432 \cdot 1 \cdot x - 432 \cdot 36 x^{2} x

432 \cdot 1 \cdot x = 432 x

432 \cdot 1 \cdot x

432 \cdot 1 = 432 :

اضرب الأعداد

= 432 x

432 \cdot 36 x^{2} x = 15552 x^{3}

432 \cdot 36 x^{2} x

432 \cdot 36 = 15552 :

اضرب الأعداد

= 15552 x^{2} x

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

x^{2} x = x^{2 + 1}

= 15552 x^{2 + 1}

2 + 1 = 3 :

اجمع الأعداد

= 15552 x^{3}

= 432 x - 15552 x^{3}

= 432 x - 15552 x^{3}

= 432 x - 15552 x^{3}

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x - 15552 x^{3}

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x - 15552 x^{3}

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x - 15552 x^{3}

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x - 15552 x^{3}

حلّ:

x \approx 0.00923 \dots, x \approx 0.00923 \dots, x \approx - 3.00922 \dots, x \approx - 3.00923 \dots

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x - 15552 x^{3}

انقل

15552 x^{3}

إلى الجانب الأيسر

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x - 15552 x^{3}

للطرفين

15552 x^{3}

أضف

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1 + 15552 x^{3} = 432 x - 15552 x^{3} + 15552 x^{3}

بسّط

1296 x^{4} + 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x

1296 x^{4} + 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x

انقل

432 x

إلى الجانب الأيسر

1296 x^{4} + 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x

من الطرفين

432 x

اطرح

1296 x^{4} + 7776 x^{3} + 11592 x^{2} + 216 x + 1 - 432 x = 432 x - 432 x

بسّط

1296 x^{4} + 7776 x^{3} + 11592 x^{2} - 216 x + 1 = 0

1296 x^{4} + 7776 x^{3} + 11592 x^{2} - 216 x + 1 = 0

1296

اقسم الطرفين على

\frac{1296 x ^{4}}{1296} + \frac{7776 x ^{3}}{1296} + \frac{11592 x ^{2}}{1296} - \frac{216 x}{1296} + \frac{1}{1296} = \frac{0}{1296}

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

اكتب بالصورة الاعتياديّة

x^{4} + 6 x^{3} + \frac{161 x ^{2}}{18} - \frac{x}{6} + \frac{1}{1296} = 0

بطريقة نيوتون ريبسون

x^{4} + 6 x^{3} + 8.94444 \dots x^{2} - 0.16666 \dots x + 0.00077 \dots = 0

جدّ حلًا لـ:

x \approx 0.00923 \dots

x^{4} + 6 x^{3} + 8.94444 \dots x^{2} - 0.16666 \dots x + 0.00077 \dots = 0

تعريف تقريب نيوتن-ريبسون

f (x) = x^{4} + 6 x^{3} + 8.94444 \dots x^{2} - 0.16666 \dots x + 0.00077 \dots

f^{^{'}} (x)

جد:

4 x^{3} + 18 x^{2} + 17.88888 \dots x - 0.16666 \dots

\frac{d}{d x} (x^{4} + 6 x^{3} + 8.94444 \dots x^{2} - 0.16666 \dots x + 0.00077 \dots)

(f \pm g)^{'} = f^{'} \pm g^{'}

:استعمل قانون الجمع

= \frac{d}{d x} (x^{4}) + \frac{d}{d x} (6 x^{3}) + \frac{d}{d x} (8.94444 \dots x^{2}) - \frac{d}{d x} (0.16666 \dots x) + \frac{d}{d x} (0.00077 \dots)

\frac{d}{d x} (x^{4}) = 4 x^{3}

\frac{d}{d x} (x^{4})

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

:استعمل قانون الأسس

= 4 x^{4 - 1}

بسّط

= 4 x^{3}

\frac{d}{d x} (6 x^{3}) = 18 x^{2}

\frac{d}{d x} (6 x^{3})

(a \cdot f)^{'} = a \cdot f^{'}

:استخرج الثابت

= 6 \frac{d}{d x} (x^{3})

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

:استعمل قانون الأسس

= 6 \cdot 3 x^{3 - 1}

بسّط

= 18 x^{2}

\frac{d}{d x} (8.94444 \dots x^{2}) = 17.88888 \dots x

\frac{d}{d x} (8.94444 \dots x^{2})

(a \cdot f)^{'} = a \cdot f^{'}

:استخرج الثابت

= 8.94444 \dots \frac{d}{d x} (x^{2})

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

:استعمل قانون الأسس

= 8.94444 \dots \cdot 2 x^{2 - 1}

بسّط

= 17.88888 \dots x

\frac{d}{d x} (0.16666 \dots x) = 0.16666 \dots

\frac{d}{d x} (0.16666 \dots x)

(a \cdot f)^{'} = a \cdot f^{'}

:استخرج الثابت

= 0.16666 \dots \frac{d x}{d x}

\frac{d x}{d x} = 1

:استعمل المشتقة الأساسية

= 0.16666 \dots \cdot 1

بسّط

= 0.16666 \dots

\frac{d}{d x} (0.00077 \dots) = 0

\frac{d}{d x} (0.00077 \dots)

\frac{d}{d x} (a) = 0

:مشتقة الثابت

= 0

= 4 x^{3} + 18 x^{2} + 17.88888 \dots x - 0.16666 \dots + 0

بسّط

= 4 x^{3} + 18 x^{2} + 17.88888 \dots x - 0.16666 \dots

x_{0} = 0

استبدل

Δ x_{n + 1} < 0.000001

حتّى

x_{n + 1}

احسب

x_{1} = 0.00462 \dots : Δ x_{1} = 0.00462 \dots

f (x_{0}) = 0^{4} + 6 \cdot 0^{3} + 8.94444 \dots \cdot 0^{2} - 0.16666 \dots \cdot 0 + 0.00077 \dots = 0.00077 \dots

f^{^{'}} (x_{0}) = 4 \cdot 0^{3} + 18 \cdot 0^{2} + 17.88888 \dots \cdot 0 - 0.16666 \dots = - 0.16666 \dots

x_{1} = 0.00462 \dots

Δ x_{1} = ∣ 0.00462 \dots - 0 ∣ = 0.00462 \dots

Δ x_{1} = 0.00462 \dots

x_{2} = 0.00693 \dots : Δ x_{2} = 0.00230 \dots

f (x_{1}) = 0.00462 \dots^{4} + 6 \cdot 0.00462 \dots^{3} + 8.94444 \dots \cdot 0.00462 \dots^{2} - 0.16666 \dots \cdot 0.00462 \dots + 0.00077 \dots = 0.00019 \dots

f^{^{'}} (x_{1}) = 4 \cdot 0.00462 \dots^{3} + 18 \cdot 0.00462 \dots^{2} + 17.88888 \dots \cdot 0.00462 \dots - 0.16666 \dots = - 0.08346 \dots

x_{2} = 0.00693 \dots

Δ x_{2} = ∣ 0.00693 \dots - 0.00462 \dots ∣ = 0.00230 \dots

Δ x_{2} = 0.00230 \dots

x_{3} = 0.00808 \dots : Δ x_{3} = 0.00114 \dots

f (x_{2}) = 0.00693 \dots^{4} + 6 \cdot 0.00693 \dots^{3} + 8.94444 \dots \cdot 0.00693 \dots^{2} - 0.16666 \dots \cdot 0.00693 \dots + 0.00077 \dots = 0.00004 \dots

f^{^{'}} (x_{2}) = 4 \cdot 0.00693 \dots^{3} + 18 \cdot 0.00693 \dots^{2} + 17.88888 \dots \cdot 0.00693 \dots - 0.16666 \dots = - 0.04176 \dots

x_{3} = 0.00808 \dots

Δ x_{3} = ∣ 0.00808 \dots - 0.00693 \dots ∣ = 0.00114 \dots

Δ x_{3} = 0.00114 \dots

x_{4} = 0.00865 \dots : Δ x_{4} = 0.00057 \dots

f (x_{3}) = 0.00808 \dots^{4} + 6 \cdot 0.00808 \dots^{3} + 8.94444 \dots \cdot 0.00808 \dots^{2} - 0.16666 \dots \cdot 0.00808 \dots + 0.00077 \dots = 0.00001 \dots

f^{^{'}} (x_{3}) = 4 \cdot 0.00808 \dots^{3} + 18 \cdot 0.00808 \dots^{2} + 17.88888 \dots \cdot 0.00808 \dots - 0.16666 \dots = - 0.02088 \dots

x_{4} = 0.00865 \dots

Δ x_{4} = ∣ 0.00865 \dots - 0.00808 \dots ∣ = 0.00057 \dots

Δ x_{4} = 0.00057 \dots

x_{5} = 0.00894 \dots : Δ x_{5} = 0.00028 \dots

f (x_{4}) = 0.00865 \dots^{4} + 6 \cdot 0.00865 \dots^{3} + 8.94444 \dots \cdot 0.00865 \dots^{2} - 0.16666 \dots \cdot 0.00865 \dots + 0.00077 \dots = 2.99676 E - 6

f^{^{'}} (x_{4}) = 4 \cdot 0.00865 \dots^{3} + 18 \cdot 0.00865 \dots^{2} + 17.88888 \dots \cdot 0.00865 \dots - 0.16666 \dots = - 0.01044 \dots

x_{5} = 0.00894 \dots

Δ x_{5} = ∣ 0.00894 \dots - 0.00865 \dots ∣ = 0.00028 \dots

Δ x_{5} = 0.00028 \dots

x_{6} = 0.00908 \dots : Δ x_{6} = 0.00014 \dots

f (x_{5}) = 0.00894 \dots^{4} + 6 \cdot 0.00894 \dots^{3} + 8.94444 \dots \cdot 0.00894 \dots^{2} - 0.16666 \dots \cdot 0.00894 \dots + 0.00077 \dots = 7.49047 E - 7

f^{^{'}} (x_{5}) = 4 \cdot 0.00894 \dots^{3} + 18 \cdot 0.00894 \dots^{2} + 17.88888 \dots \cdot 0.00894 \dots - 0.16666 \dots = - 0.00522 \dots

x_{6} = 0.00908 \dots

Δ x_{6} = ∣ 0.00908 \dots - 0.00894 \dots ∣ = 0.00014 \dots

Δ x_{6} = 0.00014 \dots

x_{7} = 0.00915 \dots : Δ x_{7} = 0.00007 \dots

f (x_{6}) = 0.00908 \dots^{4} + 6 \cdot 0.00908 \dots^{3} + 8.94444 \dots \cdot 0.00908 \dots^{2} - 0.16666 \dots \cdot 0.00908 \dots + 0.00077 \dots = 1.87244 E - 7

f^{^{'}} (x_{6}) = 4 \cdot 0.00908 \dots^{3} + 18 \cdot 0.00908 \dots^{2} + 17.88888 \dots \cdot 0.00908 \dots - 0.16666 \dots = - 0.00261 \dots

x_{7} = 0.00915 \dots

Δ x_{7} = ∣ 0.00915 \dots - 0.00908 \dots ∣ = 0.00007 \dots

Δ x_{7} = 0.00007 \dots

x_{8} = 0.00919 \dots : Δ x_{8} = 0.00003 \dots

f (x_{7}) = 0.00915 \dots^{4} + 6 \cdot 0.00915 \dots^{3} + 8.94444 \dots \cdot 0.00915 \dots^{2} - 0.16666 \dots \cdot 0.00915 \dots + 0.00077 \dots = 4.68088 E - 8

f^{^{'}} (x_{7}) = 4 \cdot 0.00915 \dots^{3} + 18 \cdot 0.00915 \dots^{2} + 17.88888 \dots \cdot 0.00915 \dots - 0.16666 \dots = - 0.00130 \dots

x_{8} = 0.00919 \dots

Δ x_{8} = ∣ 0.00919 \dots - 0.00915 \dots ∣ = 0.00003 \dots

Δ x_{8} = 0.00003 \dots

x_{9} = 0.00921 \dots : Δ x_{9} = 0.00001 \dots

f (x_{8}) = 0.00919 \dots^{4} + 6 \cdot 0.00919 \dots^{3} + 8.94444 \dots \cdot 0.00919 \dots^{2} - 0.16666 \dots \cdot 0.00919 \dots + 0.00077 \dots = 1.17019 E - 8

f^{^{'}} (x_{8}) = 4 \cdot 0.00919 \dots^{3} + 18 \cdot 0.00919 \dots^{2} + 17.88888 \dots \cdot 0.00919 \dots - 0.16666 \dots = - 0.00065 \dots

x_{9} = 0.00921 \dots

Δ x_{9} = ∣ 0.00921 \dots - 0.00919 \dots ∣ = 0.00001 \dots

Δ x_{9} = 0.00001 \dots

x_{10} = 0.00922 \dots : Δ x_{10} = 8.95953 E - 6

f (x_{9}) = 0.00921 \dots^{4} + 6 \cdot 0.00921 \dots^{3} + 8.94444 \dots \cdot 0.00921 \dots^{2} - 0.16666 \dots \cdot 0.00921 \dots + 0.00077 \dots = 2.92544 E - 9

f^{^{'}} (x_{9}) = 4 \cdot 0.00921 \dots^{3} + 18 \cdot 0.00921 \dots^{2} + 17.88888 \dots \cdot 0.00921 \dots - 0.16666 \dots = - 0.00032 \dots

x_{10} = 0.00922 \dots

Δ x_{10} = ∣ 0.00922 \dots - 0.00921 \dots ∣ = 8.95953 E - 6

Δ x_{10} = 8.95953 E - 6

x_{11} = 0.00922 \dots : Δ x_{11} = 4.47973 E - 6

f (x_{10}) = 0.00922 \dots^{4} + 6 \cdot 0.00922 \dots^{3} + 8.94444 \dots \cdot 0.00922 \dots^{2} - 0.16666 \dots \cdot 0.00922 \dots + 0.00077 \dots = 7.31357 E - 10

f^{^{'}} (x_{10}) = 4 \cdot 0.00922 \dots^{3} + 18 \cdot 0.00922 \dots^{2} + 17.88888 \dots \cdot 0.00922 \dots - 0.16666 \dots = - 0.00016 \dots

x_{11} = 0.00922 \dots

Δ x_{11} = ∣ 0.00922 \dots - 0.00922 \dots ∣ = 4.47973 E - 6

Δ x_{11} = 4.47973 E - 6

x_{12} = 0.00922 \dots : Δ x_{12} = 2.23985 E - 6

f (x_{11}) = 0.00922 \dots^{4} + 6 \cdot 0.00922 \dots^{3} + 8.94444 \dots \cdot 0.00922 \dots^{2} - 0.16666 \dots \cdot 0.00922 \dots + 0.00077 \dots = 1.82839 E - 10

f^{^{'}} (x_{11}) = 4 \cdot 0.00922 \dots^{3} + 18 \cdot 0.00922 \dots^{2} + 17.88888 \dots \cdot 0.00922 \dots - 0.16666 \dots = - 0.00008 \dots

x_{12} = 0.00922 \dots

Δ x_{12} = ∣ 0.00922 \dots - 0.00922 \dots ∣ = 2.23985 E - 6

Δ x_{12} = 2.23985 E - 6

x_{13} = 0.00922 \dots : Δ x_{13} = 1.11992 E - 6

f (x_{12}) = 0.00922 \dots^{4} + 6 \cdot 0.00922 \dots^{3} + 8.94444 \dots \cdot 0.00922 \dots^{2} - 0.16666 \dots \cdot 0.00922 \dots + 0.00077 \dots = 4.57096 E - 11

f^{^{'}} (x_{12}) = 4 \cdot 0.00922 \dots^{3} + 18 \cdot 0.00922 \dots^{2} + 17.88888 \dots \cdot 0.00922 \dots - 0.16666 \dots = - 0.00004 \dots

x_{13} = 0.00922 \dots

Δ x_{13} = ∣ 0.00922 \dots - 0.00922 \dots ∣ = 1.11992 E - 6

Δ x_{13} = 1.11992 E - 6

x_{14} = 0.00923 \dots : Δ x_{14} = 5.59961 E - 7

f (x_{13}) = 0.00922 \dots^{4} + 6 \cdot 0.00922 \dots^{3} + 8.94444 \dots \cdot 0.00922 \dots^{2} - 0.16666 \dots \cdot 0.00922 \dots + 0.00077 \dots = 1.14274 E - 11

f^{^{'}} (x_{13}) = 4 \cdot 0.00922 \dots^{3} + 18 \cdot 0.00922 \dots^{2} + 17.88888 \dots \cdot 0.00922 \dots - 0.16666 \dots = - 0.00002 \dots

x_{14} = 0.00923 \dots

Δ x_{14} = ∣ 0.00923 \dots - 0.00922 \dots ∣ = 5.59961 E - 7

Δ x_{14} = 5.59961 E - 7

x \approx 0.00923 \dots

فعّل القسمة الطويلة:

\frac{x ^{4} + 6 x ^{3} + \frac{161 x ^{2}}{18} - \frac{x}{6} + \frac{1}{1296}}{x - 0.00923 \dots} = x^{3} + 6.00923 \dots x^{2} + 8.99991 \dots x - 0.08359 \dots

x^{3} + 6.00923 \dots x^{2} + 8.99991 \dots x - 0.08359 \dots \approx 0

بطريقة نيوتون ريبسون

x^{3} + 6.00923 \dots x^{2} + 8.99991 \dots x - 0.08359 \dots = 0

جدّ حلًا لـ:

x \approx 0.00923 \dots

x^{3} + 6.00923 \dots x^{2} + 8.99991 \dots x - 0.08359 \dots = 0

تعريف تقريب نيوتن-ريبسون

f (x) = x^{3} + 6.00923 \dots x^{2} + 8.99991 \dots x - 0.08359 \dots

f^{^{'}} (x)

جد:

3 x^{2} + 12.01846 \dots x + 8.99991 \dots

\frac{d}{d x} (x^{3} + 6.00923 \dots x^{2} + 8.99991 \dots x - 0.08359 \dots)

(f \pm g)^{'} = f^{'} \pm g^{'}

:استعمل قانون الجمع

= \frac{d}{d x} (x^{3}) + \frac{d}{d x} (6.00923 \dots x^{2}) + \frac{d}{d x} (8.99991 \dots x) - \frac{d}{d x} (0.08359 \dots)

\frac{d}{d x} (x^{3}) = 3 x^{2}

\frac{d}{d x} (x^{3})

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

:استعمل قانون الأسس

= 3 x^{3 - 1}

بسّط

= 3 x^{2}

\frac{d}{d x} (6.00923 \dots x^{2}) = 12.01846 \dots x

\frac{d}{d x} (6.00923 \dots x^{2})

(a \cdot f)^{'} = a \cdot f^{'}

:استخرج الثابت

= 6.00923 \dots \frac{d}{d x} (x^{2})

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

:استعمل قانون الأسس

= 6.00923 \dots \cdot 2 x^{2 - 1}

بسّط

= 12.01846 \dots x

\frac{d}{d x} (8.99991 \dots x) = 8.99991 \dots

\frac{d}{d x} (8.99991 \dots x)

(a \cdot f)^{'} = a \cdot f^{'}

:استخرج الثابت

= 8.99991 \dots \frac{d x}{d x}

\frac{d x}{d x} = 1

:استعمل المشتقة الأساسية

= 8.99991 \dots \cdot 1

بسّط

= 8.99991 \dots

\frac{d}{d x} (0.08359 \dots) = 0

\frac{d}{d x} (0.08359 \dots)

\frac{d}{d x} (a) = 0

:مشتقة الثابت

= 0

= 3 x^{2} + 12.01846 \dots x + 8.99991 \dots - 0

بسّط

= 3 x^{2} + 12.01846 \dots x + 8.99991 \dots

x_{0} = 0

استبدل

Δ x_{n + 1} < 0.000001

حتّى

x_{n + 1}

احسب

x_{1} = 0.00928 \dots : Δ x_{1} = 0.00928 \dots

f (x_{0}) = 0^{3} + 6.00923 \dots \cdot 0^{2} + 8.99991 \dots \cdot 0 - 0.08359 \dots = - 0.08359 \dots

f^{^{'}} (x_{0}) = 3 \cdot 0^{2} + 12.01846 \dots \cdot 0 + 8.99991 \dots = 8.99991 \dots

x_{1} = 0.00928 \dots

Δ x_{1} = ∣ 0.00928 \dots - 0 ∣ = 0.00928 \dots

Δ x_{1} = 0.00928 \dots

x_{2} = 0.00923 \dots : Δ x_{2} = 0.00005 \dots

f (x_{1}) = 0.00928 \dots^{3} + 6.00923 \dots \cdot 0.00928 \dots^{2} + 8.99991 \dots \cdot 0.00928 \dots - 0.08359 \dots = 0.00051 \dots

f^{^{'}} (x_{1}) = 3 \cdot 0.00928 \dots^{2} + 12.01846 \dots \cdot 0.00928 \dots + 8.99991 \dots = 9.11180 \dots

x_{2} = 0.00923 \dots

Δ x_{2} = ∣ 0.00923 \dots - 0.00928 \dots ∣ = 0.00005 \dots

Δ x_{2} = 0.00005 \dots

x_{3} = 0.00923 \dots : Δ x_{3} = 2.15173 E - 9

f (x_{2}) = 0.00923 \dots^{3} + 6.00923 \dots \cdot 0.00923 \dots^{2} + 8.99991 \dots \cdot 0.00923 \dots - 0.08359 \dots = 1.96046 E - 8

f^{^{'}} (x_{2}) = 3 \cdot 0.00923 \dots^{2} + 12.01846 \dots \cdot 0.00923 \dots + 8.99991 \dots = 9.11111 \dots

x_{3} = 0.00923 \dots

Δ x_{3} = ∣ 0.00923 \dots - 0.00923 \dots ∣ = 2.15173 E - 9

Δ x_{3} = 2.15173 E - 9

x \approx 0.00923 \dots

فعّل القسمة الطويلة:

\frac{x ^{3} + 6.00923 \dots x ^{2} + 8.99991 \dots x - 0.08359 \dots}{x - 0.00923 \dots} = x^{2} + 6.01846 \dots x + 9.05547 \dots

x^{2} + 6.01846 \dots x + 9.05547 \dots \approx 0

بطريقة نيوتون ريبسون

x^{2} + 6.01846 \dots x + 9.05547 \dots = 0

جدّ حلًا لـ:

x \approx - 3.00922 \dots

x^{2} + 6.01846 \dots x + 9.05547 \dots = 0

تعريف تقريب نيوتن-ريبسون

f (x) = x^{2} + 6.01846 \dots x + 9.05547 \dots

f^{^{'}} (x)

جد:

2 x + 6.01846 \dots

\frac{d}{d x} (x^{2} + 6.01846 \dots x + 9.05547 \dots)

(f \pm g)^{'} = f^{'} \pm g^{'}

:استعمل قانون الجمع

= \frac{d}{d x} (x^{2}) + \frac{d}{d x} (6.01846 \dots x) + \frac{d}{d x} (9.05547 \dots)

\frac{d}{d x} (x^{2}) = 2 x

\frac{d}{d x} (x^{2})

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

:استعمل قانون الأسس

= 2 x^{2 - 1}

بسّط

= 2 x

\frac{d}{d x} (6.01846 \dots x) = 6.01846 \dots

\frac{d}{d x} (6.01846 \dots x)

(a \cdot f)^{'} = a \cdot f^{'}

:استخرج الثابت

= 6.01846 \dots \frac{d x}{d x}

\frac{d x}{d x} = 1

:استعمل المشتقة الأساسية

= 6.01846 \dots \cdot 1

بسّط

= 6.01846 \dots

\frac{d}{d x} (9.05547 \dots) = 0

\frac{d}{d x} (9.05547 \dots)

\frac{d}{d x} (a) = 0

:مشتقة الثابت

= 0

= 2 x + 6.01846 \dots + 0

بسّط

= 2 x + 6.01846 \dots

x_{0} = - 2

استبدل

Δ x_{n + 1} < 0.000001

حتّى

x_{n + 1}

احسب

x_{1} = - 2.50461 \dots : Δ x_{1} = 0.50461 \dots

f (x_{0}) = (- 2)^{2} + 6.01846 \dots (- 2) + 9.05547 \dots = 1.01854 \dots

f^{^{'}} (x_{0}) = 2 (- 2) + 6.01846 \dots = 2.01846 \dots

x_{1} = - 2.50461 \dots

Δ x_{1} = ∣ - 2.50461 \dots - (- 2) ∣ = 0.50461 \dots

Δ x_{1} = 0.50461 \dots

x_{2} = - 2.75692 \dots : Δ x_{2} = 0.25230 \dots

f (x_{1}) = (- 2.50461 \dots)^{2} + 6.01846 \dots (- 2.50461 \dots) + 9.05547 \dots = 0.25463 \dots

f^{^{'}} (x_{1}) = 2 (- 2.50461 \dots) + 6.01846 \dots = 1.00923 \dots

x_{2} = - 2.75692 \dots

Δ x_{2} = ∣ - 2.75692 \dots - (- 2.50461 \dots) ∣ = 0.25230 \dots

Δ x_{2} = 0.25230 \dots

x_{3} = - 2.88307 \dots : Δ x_{3} = 0.12615 \dots

f (x_{2}) = (- 2.75692 \dots)^{2} + 6.01846 \dots (- 2.75692 \dots) + 9.05547 \dots = 0.06365 \dots

f^{^{'}} (x_{2}) = 2 (- 2.75692 \dots) + 6.01846 \dots = 0.50461 \dots

x_{3} = - 2.88307 \dots

Δ x_{3} = ∣ - 2.88307 \dots - (- 2.75692 \dots) ∣ = 0.12615 \dots

Δ x_{3} = 0.12615 \dots

x_{4} = - 2.94615 \dots : Δ x_{4} = 0.06307 \dots

f (x_{3}) = (- 2.88307 \dots)^{2} + 6.01846 \dots (- 2.88307 \dots) + 9.05547 \dots = 0.01591 \dots

f^{^{'}} (x_{3}) = 2 (- 2.88307 \dots) + 6.01846 \dots = 0.25230 \dots

x_{4} = - 2.94615 \dots

Δ x_{4} = ∣ - 2.94615 \dots - (- 2.88307 \dots) ∣ = 0.06307 \dots

Δ x_{4} = 0.06307 \dots

x_{5} = - 2.97769 \dots : Δ x_{5} = 0.03153 \dots

f (x_{4}) = (- 2.94615 \dots)^{2} + 6.01846 \dots (- 2.94615 \dots) + 9.05547 \dots = 0.00397 \dots

f^{^{'}} (x_{4}) = 2 (- 2.94615 \dots) + 6.01846 \dots = 0.12615 \dots

x_{5} = - 2.97769 \dots

Δ x_{5} = ∣ - 2.97769 \dots - (- 2.94615 \dots) ∣ = 0.03153 \dots

Δ x_{5} = 0.03153 \dots

x_{6} = - 2.99346 \dots : Δ x_{6} = 0.01576 \dots

f (x_{5}) = (- 2.97769 \dots)^{2} + 6.01846 \dots (- 2.97769 \dots) + 9.05547 \dots = 0.00099 \dots

f^{^{'}} (x_{5}) = 2 (- 2.97769 \dots) + 6.01846 \dots = 0.06307 \dots

x_{6} = - 2.99346 \dots

Δ x_{6} = ∣ - 2.99346 \dots - (- 2.97769 \dots) ∣ = 0.01576 \dots

Δ x_{6} = 0.01576 \dots

x_{7} = - 3.00134 \dots : Δ x_{7} = 0.00788 \dots

f (x_{6}) = (- 2.99346 \dots)^{2} + 6.01846 \dots (- 2.99346 \dots) + 9.05547 \dots = 0.00024 \dots

f^{^{'}} (x_{6}) = 2 (- 2.99346 \dots) + 6.01846 \dots = 0.03153 \dots

x_{7} = - 3.00134 \dots

Δ x_{7} = ∣ - 3.00134 \dots - (- 2.99346 \dots) ∣ = 0.00788 \dots

Δ x_{7} = 0.00788 \dots

x_{8} = - 3.00528 \dots : Δ x_{8} = 0.00394 \dots

f (x_{7}) = (- 3.00134 \dots)^{2} + 6.01846 \dots (- 3.00134 \dots) + 9.05547 \dots = 0.00006 \dots

f^{^{'}} (x_{7}) = 2 (- 3.00134 \dots) + 6.01846 \dots = 0.01576 \dots

x_{8} = - 3.00528 \dots

Δ x_{8} = ∣ - 3.00528 \dots - (- 3.00134 \dots) ∣ = 0.00394 \dots

Δ x_{8} = 0.00394 \dots

x_{9} = - 3.00725 \dots : Δ x_{9} = 0.00197 \dots

f (x_{8}) = (- 3.00528 \dots)^{2} + 6.01846 \dots (- 3.00528 \dots) + 9.05547 \dots = 0.00001 \dots

f^{^{'}} (x_{8}) = 2 (- 3.00528 \dots) + 6.01846 \dots = 0.00788 \dots

x_{9} = - 3.00725 \dots

Δ x_{9} = ∣ - 3.00725 \dots - (- 3.00528 \dots) ∣ = 0.00197 \dots

Δ x_{9} = 0.00197 \dots

x_{10} = - 3.00824 \dots : Δ x_{10} = 0.00098 \dots

f (x_{9}) = (- 3.00725 \dots)^{2} + 6.01846 \dots (- 3.00725 \dots) + 9.05547 \dots = 3.88545 E - 6

f^{^{'}} (x_{9}) = 2 (- 3.00725 \dots) + 6.01846 \dots = 0.00394 \dots

x_{10} = - 3.00824 \dots

Δ x_{10} = ∣ - 3.00824 \dots - (- 3.00725 \dots) ∣ = 0.00098 \dots

Δ x_{10} = 0.00098 \dots

x_{11} = - 3.00873 \dots : Δ x_{11} = 0.00049 \dots

f (x_{10}) = (- 3.00824 \dots)^{2} + 6.01846 \dots (- 3.00824 \dots) + 9.05547 \dots = 9.71362 E - 7

f^{^{'}} (x_{10}) = 2 (- 3.00824 \dots) + 6.01846 \dots = 0.00197 \dots

x_{11} = - 3.00873 \dots

Δ x_{11} = ∣ - 3.00873 \dots - (- 3.00824 \dots) ∣ = 0.00049 \dots

Δ x_{11} = 0.00049 \dots

x_{12} = - 3.00898 \dots : Δ x_{12} = 0.00024 \dots

f (x_{11}) = (- 3.00873 \dots)^{2} + 6.01846 \dots (- 3.00873 \dots) + 9.05547 \dots = 2.4284 E - 7

f^{^{'}} (x_{11}) = 2 (- 3.00873 \dots) + 6.01846 \dots = 0.00098 \dots

x_{12} = - 3.00898 \dots

Δ x_{12} = ∣ - 3.00898 \dots - (- 3.00873 \dots) ∣ = 0.00024 \dots

Δ x_{12} = 0.00024 \dots

x_{13} = - 3.00910 \dots : Δ x_{13} = 0.00012 \dots

f (x_{12}) = (- 3.00898 \dots)^{2} + 6.01846 \dots (- 3.00898 \dots) + 9.05547 \dots = 6.071 E - 8

f^{^{'}} (x_{12}) = 2 (- 3.00898 \dots) + 6.01846 \dots = 0.00049 \dots

x_{13} = - 3.00910 \dots

Δ x_{13} = ∣ - 3.00910 \dots - (- 3.00898 \dots) ∣ = 0.00012 \dots

Δ x_{13} = 0.00012 \dots

x_{14} = - 3.00916 \dots : Δ x_{14} = 0.00006 \dots

f (x_{13}) = (- 3.00910 \dots)^{2} + 6.01846 \dots (- 3.00910 \dots) + 9.05547 \dots = 1.51774 E - 8

f^{^{'}} (x_{13}) = 2 (- 3.00910 \dots) + 6.01846 \dots = 0.00024 \dots

x_{14} = - 3.00916 \dots

Δ x_{14} = ∣ - 3.00916 \dots - (- 3.00910 \dots) ∣ = 0.00006 \dots

Δ x_{14} = 0.00006 \dots

x_{15} = - 3.00920 \dots : Δ x_{15} = 0.00003 \dots

f (x_{14}) = (- 3.00916 \dots)^{2} + 6.01846 \dots (- 3.00916 \dots) + 9.05547 \dots = 3.79428 E - 9

f^{^{'}} (x_{14}) = 2 (- 3.00916 \dots) + 6.01846 \dots = 0.00012 \dots

x_{15} = - 3.00920 \dots

Δ x_{15} = ∣ - 3.00920 \dots - (- 3.00916 \dots) ∣ = 0.00003 \dots

Δ x_{15} = 0.00003 \dots

x_{16} = - 3.00921 \dots : Δ x_{16} = 0.00001 \dots

f (x_{15}) = (- 3.00920 \dots)^{2} + 6.01846 \dots (- 3.00920 \dots) + 9.05547 \dots = 9.48491 E - 10

f^{^{'}} (x_{15}) = 2 (- 3.00920 \dots) + 6.01846 \dots = 0.00006 \dots

x_{16} = - 3.00921 \dots

Δ x_{16} = ∣ - 3.00921 \dots - (- 3.00920 \dots) ∣ = 0.00001 \dots

Δ x_{16} = 0.00001 \dots

x_{17} = - 3.00922 \dots : Δ x_{17} = 7.69303 E - 6

f (x_{16}) = (- 3.00921 \dots)^{2} + 6.01846 \dots (- 3.00921 \dots) + 9.05547 \dots = 2.37044 E - 10

f^{^{'}} (x_{16}) = 2 (- 3.00921 \dots) + 6.01846 \dots = 0.00003 \dots

x_{17} = - 3.00922 \dots

Δ x_{17} = ∣ - 3.00922 \dots - (- 3.00921 \dots) ∣ = 7.69303 E - 6

Δ x_{17} = 7.69303 E - 6

x_{18} = - 3.00922 \dots : Δ x_{18} = 3.83637 E - 6

f (x_{17}) = (- 3.00922 \dots)^{2} + 6.01846 \dots (- 3.00922 \dots) + 9.05547 \dots = 5.91829 E - 11

f^{^{'}} (x_{17}) = 2 (- 3.00922 \dots) + 6.01846 \dots = 0.00001 \dots

x_{18} = - 3.00922 \dots

Δ x_{18} = ∣ - 3.00922 \dots - (- 3.00922 \dots) ∣ = 3.83637 E - 6

Δ x_{18} = 3.83637 E - 6

x_{19} = - 3.00922 \dots : Δ x_{19} = 1.89799 E - 6

f (x_{18}) = (- 3.00922 \dots)^{2} + 6.01846 \dots (- 3.00922 \dots) + 9.05547 \dots = 1.47171 E - 11

f^{^{'}} (x_{18}) = 2 (- 3.00922 \dots) + 6.01846 \dots = 7.75406 E - 6

x_{19} = - 3.00922 \dots

Δ x_{19} = ∣ - 3.00922 \dots - (- 3.00922 \dots) ∣ = 1.89799 E - 6

Δ x_{19} = 1.89799 E - 6

x_{20} = - 3.00922 \dots : Δ x_{20} = 9.10151 E - 7

f (x_{19}) = (- 3.00922 \dots)^{2} + 6.01846 \dots (- 3.00922 \dots) + 9.05547 \dots = 3.60245 E - 12

f^{^{'}} (x_{19}) = 2 (- 3.00922 \dots) + 6.01846 \dots = 3.95808 E - 6

x_{20} = - 3.00922 \dots

Δ x_{20} = ∣ - 3.00922 \dots - (- 3.00922 \dots) ∣ = 9.10151 E - 7

Δ x_{20} = 9.10151 E - 7

x \approx - 3.00922 \dots

فعّل القسمة الطويلة:

\frac{x ^{2} + 6.01846 \dots x + 9.05547 \dots}{x + 3.00922 \dots} = x + 3.00923 \dots

x + 3.00923 \dots \approx 0

x \approx - 3.00923 \dots

The solutions are

x \approx 0.00923 \dots, x \approx 0.00923 \dots, x \approx - 3.00922 \dots, x \approx - 3.00923 \dots

x \approx 0.00923 \dots, x \approx 0.00923 \dots, x \approx - 3.00922 \dots, x \approx - 3.00923 \dots

افحص الإجبات:

x \approx 0.00923 \dots

خطأ

, x \approx 0.00923 \dots

خطأ

, x \approx - 3.00922 \dots

خطأ

, x \approx - 3.00923 \dots

خطأ

للتحقّق من دقّة الحلول

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} = - 1

عوّض الحلول في

إلغي الحلول التي تعطي قضيّة كذب

x \approx 0.00923 \dots

استبدل:

خطأ

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2} + 6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2} = - 1

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2} + 6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2} = 0.99999 \dots

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2} + 6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2}

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2} = 0.05538 \dots 0.00312 \dots

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2}

1 - (6 3 \cdot 0.00923 \dots)^{2} = 0.00312 \dots

1 - (6 3 \cdot 0.00923 \dots)^{2}

(6 3 \cdot 0.00923 \dots)^{2} = 0.99687 \dots

(6 3 \cdot 0.00923 \dots)^{2}

3 \cdot 0.00923 \dots = 0.02769 \dots :

اضرب الأعداد

= (6 0.02769 \dots)^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= 6^{2} (0.02769 \dots)^{2}

(0.02769 \dots)^{2} : 0.02769 \dots

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

= (0.02769 \dots^{\frac{1}{2}})^{2}

(a^{b})^{c} = a^{b c}

:فعّل قانون القوى

= 0.02769 \dots^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{1 \cdot 2}{2}

2 :

إلغ العوامل المشتركة

= 1

= 0.02769 \dots

= 6^{2} \cdot 0.02769 \dots

6^{2} = 36

= 36 \cdot 0.02769 \dots

36 \cdot 0.02769 \dots = 0.99687 \dots :

اضرب الأعداد

= 0.99687 \dots

= 1 - 0.99687 \dots

1 - 0.99687 \dots = 0.00312 \dots :

اطرح الأعداد

= 0.00312 \dots

= 6 \cdot 0.00923 \dots 0.00312 \dots

6 \cdot 0.00923 \dots = 0.05538 \dots :

اضرب الأعداد

= 0.05538 \dots 0.00312 \dots

6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2} = 6 0.02760 \dots

6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2}

3 \cdot 0.00923 \dots = 0.02769 \dots :

اضرب الأعداد

= 6 0.02769 \dots - (6 \cdot 0.00923 \dots)^{2} + 1

1 - (6 \cdot 0.00923 \dots)^{2} = 0.99693 \dots

1 - (6 \cdot 0.00923 \dots)^{2}

(6 \cdot 0.00923 \dots)^{2} = 0.00306 \dots

(6 \cdot 0.00923 \dots)^{2}

6 \cdot 0.00923 \dots = 0.05538 \dots :

اضرب الأعداد

= 0.05538 \dots^{2}

0.05538 \dots^{2} = 0.00306 \dots

= 0.00306 \dots

= 1 - 0.00306 \dots

1 - 0.00306 \dots = 0.99693 \dots :

اطرح الأعداد

= 0.99693 \dots

= 6 0.02769 \dots 0.99693 \dots

a b = a \cdot b

:فعْل قانون الجذور

0.02769 \dots 0.99693 \dots = 0.02769 \dots \cdot 0.99693 \dots

= 6 0.02769 \dots \cdot 0.99693 \dots

0.02769 \dots \cdot 0.99693 \dots = 0.02760 \dots :

اضرب الأعداد

= 6 0.02760 \dots

= 0.05538 \dots 0.00312 \dots + 6 0.02760 \dots

0.05538 \dots 0.00312 \dots = 0.00309 \dots

0.05538 \dots 0.00312 \dots

0.00312 \dots = 0.05592 \dots

= 0.05538 \dots \cdot 0.05592 \dots

0.05538 \dots \cdot 0.05592 \dots = 0.00309 \dots :

اضرب الأعداد

= 0.00309 \dots

6 0.02760 \dots = 0.99690 \dots

6 0.02760 \dots

0.02760 \dots = 0.16615 \dots

= 6 \cdot 0.16615 \dots

6 \cdot 0.16615 \dots = 0.99690 \dots :

اضرب الأعداد

= 0.99690 \dots

= 0.00309 \dots + 0.99690 \dots

0.00309 \dots + 0.99690 \dots = 0.99999 \dots :

اجمع الأعداد

= 0.99999 \dots

0.99999 \dots = - 1

خطأ

x \approx 0.00923 \dots

استبدل:

خطأ

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2} + 6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2} = - 1

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2} + 6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2} = 0.99999 \dots

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2} + 6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2}

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2} = 0.05538 \dots 0.00300 \dots

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2}

1 - (6 3 \cdot 0.00923 \dots)^{2} = 0.00300 \dots

1 - (6 3 \cdot 0.00923 \dots)^{2}

(6 3 \cdot 0.00923 \dots)^{2} = 0.99699 \dots

(6 3 \cdot 0.00923 \dots)^{2}

3 \cdot 0.00923 \dots = 0.02769 \dots :

اضرب الأعداد

= (6 0.02769 \dots)^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= 6^{2} (0.02769 \dots)^{2}

(0.02769 \dots)^{2} : 0.02769 \dots

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

= (0.02769 \dots^{\frac{1}{2}})^{2}

(a^{b})^{c} = a^{b c}

:فعّل قانون القوى

= 0.02769 \dots^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{1 \cdot 2}{2}

2 :

إلغ العوامل المشتركة

= 1

= 0.02769 \dots

= 6^{2} \cdot 0.02769 \dots

6^{2} = 36

= 36 \cdot 0.02769 \dots

36 \cdot 0.02769 \dots = 0.99699 \dots :

اضرب الأعداد

= 0.99699 \dots

= 1 - 0.99699 \dots

1 - 0.99699 \dots = 0.00300 \dots :

اطرح الأعداد

= 0.00300 \dots

= 6 \cdot 0.00923 \dots 0.00300 \dots

6 \cdot 0.00923 \dots = 0.05538 \dots :

اضرب الأعداد

= 0.05538 \dots 0.00300 \dots

6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2} = 6 0.02760 \dots

6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2}

3 \cdot 0.00923 \dots = 0.02769 \dots :

اضرب الأعداد

= 6 0.02769 \dots - (6 \cdot 0.00923 \dots)^{2} + 1

1 - (6 \cdot 0.00923 \dots)^{2} = 0.99693 \dots

1 - (6 \cdot 0.00923 \dots)^{2}

(6 \cdot 0.00923 \dots)^{2} = 0.00306 \dots

(6 \cdot 0.00923 \dots)^{2}

6 \cdot 0.00923 \dots = 0.05538 \dots :

اضرب الأعداد

= 0.05538 \dots^{2}

0.05538 \dots^{2} = 0.00306 \dots

= 0.00306 \dots

= 1 - 0.00306 \dots

1 - 0.00306 \dots = 0.99693 \dots :

اطرح الأعداد

= 0.99693 \dots

= 6 0.02769 \dots 0.99693 \dots

a b = a \cdot b

:فعْل قانون الجذور

0.02769 \dots 0.99693 \dots = 0.02769 \dots \cdot 0.99693 \dots

= 6 0.02769 \dots \cdot 0.99693 \dots

0.02769 \dots \cdot 0.99693 \dots = 0.02760 \dots :

اضرب الأعداد

= 6 0.02760 \dots

= 0.05538 \dots 0.00300 \dots + 6 0.02760 \dots

0.05538 \dots 0.00300 \dots = 0.00303 \dots

0.05538 \dots 0.00300 \dots

0.00300 \dots = 0.05483 \dots

= 0.05538 \dots \cdot 0.05483 \dots

0.05538 \dots \cdot 0.05483 \dots = 0.00303 \dots :

اضرب الأعداد

= 0.00303 \dots

6 0.02760 \dots = 0.99696 \dots

6 0.02760 \dots

0.02760 \dots = 0.16616 \dots

= 6 \cdot 0.16616 \dots

6 \cdot 0.16616 \dots = 0.99696 \dots :

اضرب الأعداد

= 0.99696 \dots

= 0.00303 \dots + 0.99696 \dots

0.00303 \dots + 0.99696 \dots = 0.99999 \dots :

اجمع الأعداد

= 0.99999 \dots

0.99999 \dots = - 1

خطأ

x \approx - 3.00922 \dots

استبدل:

خطأ

6 (- 3.00922 \dots) 1 - (6 3 (- 3.00922 \dots))^{2} + 6 3 (- 3.00922 \dots) 1 - (6 (- 3.00922 \dots))^{2} = - 1

6 (- 3.00922 \dots) 1 - (6 3 (- 3.00922 \dots))^{2} + 6 3 (- 3.00922 \dots) 1 - (6 (- 3.00922 \dots))^{2} =

غير معرّف

= - 1

خطأ

x \approx - 3.00923 \dots

استبدل:

خطأ

6 (- 3.00923 \dots) 1 - (6 3 (- 3.00923 \dots))^{2} + 6 3 (- 3.00923 \dots) 1 - (6 (- 3.00923 \dots))^{2} = - 1

6 (- 3.00923 \dots) 1 - (6 3 (- 3.00923 \dots))^{2} + 6 3 (- 3.00923 \dots) 1 - (6 (- 3.00923 \dots))^{2} =

غير معرّف

= - 1

خطأ

الحل للمعادلة هو

x \in R لا يوجد حلّ لـ

لا يوجد حلّ

تأكّد من صحّة الحلول عن طريق تعويضها في المعادلة الأصليّة

للتحقّق من دقّة الحلول

arcsin (6 x) + arcsin (6 3 x) = - \frac{π}{2}

عوّض الحلول في

إلغي الحلول التي تعطي قضيّة كذب

x \in R لا يوجد حلّ لـ

رسم

أمثلة شائعة

2cos^2(x)+sin(x)=5

2 cos^{2} (x) + sin (x) = 5

solvefor x,Y=0.5sin(3.07x-2.4t+0.59)

so l v e f or x, Y = 0.5 sin (3.07 x - 2.4 t + 0.59)

sin(x-30)cos(x-30)=(sqrt(3))/4

sin (x - 3 0^{\circ}) cos (x - 3 0^{\circ}) = \frac{3}{4}

cos(θ)=-7/15 ,cos(θ/2),180<θ<270

cos (θ) = - \frac{7}{15}, cos (\frac{θ}{2}), 18 0^{\circ} < θ < 27 0^{\circ}

tan(x/2)=4

tan (\frac{x}{2}) = 4