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受欢迎的三角函数 >

arcsin(x)+arcsin(2x)= pi/3

解答

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

解答

x = \frac{3}{2 7}

求解步骤

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

使用三角恒等式改写

arcsin (x) + arcsin (2 x)

使用和差化积恒等式:

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

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

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

使用反三角函数性质

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

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

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

sin (\frac{π}{3}) = \frac{3}{2}

sin (\frac{π}{3})

使用以下普通恒等式:

sin (\frac{π}{3}) = \frac{3}{2}

sin (\frac{π}{3})

sin (x)

周期表（周期为

2 πn

"）：

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}

= \frac{3}{2}

= \frac{3}{2}

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

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

解

x 1 - (2 x)^{2} + 2 x 1 - x^{2} = \frac{3}{2} : x = \frac{1}{2}, x = \frac{3}{2 7}

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

在两边乘以

2

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

化简

2 1 - (2 x)^{2} x + 4 1 - x^{2} x = 3

去除平方根

2 1 - (2 x)^{2} x + 4 1 - x^{2} x = 3

两边减去

4 1 - x^{2} x

2 1 - (2 x)^{2} x + 4 1 - x^{2} x - 4 1 - x^{2} x = 3 - 4 1 - x^{2} x

化简

2 1 - (2 x)^{2} x = 3 - 4 1 - x^{2} x

两边进行平方:

4 x^{2} - 16 x^{4} = 3 - 83 x 1 - x^{2} + 16 x^{2} - 16 x^{4}

2 1 - (2 x)^{2} x + 4 1 - x^{2} x = 3

(2 1 - (2 x)^{2} x)^{2} = (3 - 4 1 - x^{2} x)^{2}

展开

(2 1 - (2 x)^{2} x)^{2} : 4 x^{2} - 16 x^{4}

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

使用指数法则:

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

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

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

使用根式运算法则:

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

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

使用指数法则:

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

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

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= 1 - (2 x)^{2}

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

2^{2} = 4

= 4 (1 - (2 x)^{2}) x^{2}

展开

4 (1 - (2 x)^{2}) x^{2} : 4 x^{2} - 16 x^{4}

4 (1 - (2 x)^{2}) x^{2}

使用指数法则:

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

= 4 x^{2} (- 2^{2} x^{2} + 1)

2^{2} = 4

= 4 x^{2} (- 4 x^{2} + 1)

= 4 x^{2} (1 - 4 x^{2})

使用分配律:

a (b - c) = ab - a c

a = 4 x^{2}, b = 1, c = 4 x^{2}

= 4 x^{2} \cdot 1 - 4 x^{2} \cdot 4 x^{2}

= 4 \cdot 1 \cdot x^{2} - 4 \cdot 4 x^{2} x^{2}

化简

4 \cdot 1 \cdot x^{2} - 4 \cdot 4 x^{2} x^{2} : 4 x^{2} - 16 x^{4}

4 \cdot 1 \cdot x^{2} - 4 \cdot 4 x^{2} x^{2}

4 \cdot 1 \cdot x^{2} = 4 x^{2}

4 \cdot 1 \cdot x^{2}

数字相乘:

4 \cdot 1 = 4

= 4 x^{2}

4 \cdot 4 x^{2} x^{2} = 16 x^{4}

4 \cdot 4 x^{2} x^{2}

数字相乘:

4 \cdot 4 = 16

= 16 x^{2} x^{2}

使用指数法则:

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

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

= 16 x^{2 + 2}

数字相加:

2 + 2 = 4

= 16 x^{4}

= 4 x^{2} - 16 x^{4}

= 4 x^{2} - 16 x^{4}

= 4 x^{2} - 16 x^{4}

展开

(3 - 4 1 - x^{2} x)^{2} : 3 - 83 x 1 - x^{2} + 16 x^{2} - 16 x^{4}

(3 - 4 1 - x^{2} x)^{2}

使用完全平方公式:

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

a = 3, b = 4 1 - x^{2} x

= (3)^{2} - 23 \cdot 4 1 - x^{2} x + (4 1 - x^{2} x)^{2}

化简

(3)^{2} - 23 \cdot 4 1 - x^{2} x + (4 1 - x^{2} x)^{2} : 3 - 83 1 - x^{2} x + 161 - x^{2} x^{2}

(3)^{2} - 23 \cdot 4 1 - x^{2} x + (4 1 - x^{2} x)^{2}

(3)^{2} = 3

(3)^{2}

使用根式运算法则:

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

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

使用指数法则:

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

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

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= 3

23 \cdot 4 1 - x^{2} x = 83 1 - x^{2} x

23 \cdot 4 1 - x^{2} x

数字相乘:

2 \cdot 4 = 8

= 83 1 - x^{2} x

(4 1 - x^{2} x)^{2} = 161 - x^{2} x^{2}

(4 1 - x^{2} x)^{2}

使用指数法则:

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

= 4^{2} x^{2} (1 - x^{2})^{2}

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

使用根式运算法则:

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

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

使用指数法则:

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

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

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= 1 - x^{2}

= 4^{2} (1 - x^{2}) x^{2}

4^{2} = 16

= 16 (1 - x^{2}) x^{2}

= 3 - 83 1 - x^{2} x + 16 (1 - x^{2}) x^{2}

= 3 - 83 1 - x^{2} x + 16 (1 - x^{2}) x^{2}

展开

3 - 83 1 - x^{2} x + 16 (1 - x^{2}) x^{2} : 3 - 83 x 1 - x^{2} + 16 x^{2} - 16 x^{4}

3 - 83 1 - x^{2} x + 16 (1 - x^{2}) x^{2}

= 3 - 83 x 1 - x^{2} + 16 x^{2} (1 - x^{2})

乘开

16 x^{2} (1 - x^{2}) : 16 x^{2} - 16 x^{4}

16 x^{2} (1 - x^{2})

使用分配律:

a (b - c) = ab - a c

a = 16 x^{2}, b = 1, c = x^{2}

= 16 x^{2} \cdot 1 - 16 x^{2} x^{2}

= 16 \cdot 1 \cdot x^{2} - 16 x^{2} x^{2}

化简

16 \cdot 1 \cdot x^{2} - 16 x^{2} x^{2} : 16 x^{2} - 16 x^{4}

16 \cdot 1 \cdot x^{2} - 16 x^{2} x^{2}

16 \cdot 1 \cdot x^{2} = 16 x^{2}

16 \cdot 1 \cdot x^{2}

数字相乘:

16 \cdot 1 = 16

= 16 x^{2}

16 x^{2} x^{2} = 16 x^{4}

16 x^{2} x^{2}

使用指数法则:

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

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

= 16 x^{2 + 2}

数字相加:

2 + 2 = 4

= 16 x^{4}

= 16 x^{2} - 16 x^{4}

= 16 x^{2} - 16 x^{4}

= 3 - 83 1 - x^{2} x + 16 x^{2} - 16 x^{4}

= 3 - 83 x 1 - x^{2} + 16 x^{2} - 16 x^{4}

= 3 - 83 x 1 - x^{2} + 16 x^{2} - 16 x^{4}

4 x^{2} - 16 x^{4} = 3 - 83 x 1 - x^{2} + 16 x^{2} - 16 x^{4}

4 x^{2} - 16 x^{4} = 3 - 83 x 1 - x^{2} + 16 x^{2} - 16 x^{4}

两边减去

16 x^{2} - 16 x^{4}

4 x^{2} - 16 x^{4} - (16 x^{2} - 16 x^{4}) = 3 - 83 x 1 - x^{2} + 16 x^{2} - 16 x^{4} - (16 x^{2} - 16 x^{4})

化简

- 12 x^{2} = - 83 1 - x^{2} x + 3

两边减去

3

- 12 x^{2} - 3 = - 83 1 - x^{2} x + 3 - 3

化简

- 12 x^{2} - 3 = - 83 1 - x^{2} x

两边进行平方:

144 x^{4} + 72 x^{2} + 9 = 192 x^{2} - 192 x^{4}

4 x^{2} - 16 x^{4} = 3 - 83 x 1 - x^{2} + 16 x^{2} - 16 x^{4}

(- 12 x^{2} - 3)^{2} = (- 83 1 - x^{2} x)^{2}

展开

(- 12 x^{2} - 3)^{2} : 144 x^{4} + 72 x^{2} + 9

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

使用完全平方公式:

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

a = - 12 x^{2}, b = 3

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

化简

(- 12 x^{2})^{2} - 2 (- 12 x^{2}) \cdot 3 + 3^{2} : 144 x^{4} + 72 x^{2} + 9

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

使用法则

- (- a) = a

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

(- 12 x^{2})^{2} = 144 x^{4}

(- 12 x^{2})^{2}

使用指数法则:

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

若

n

是偶数

(- 12 x^{2})^{2} = (12 x^{2})^{2}

= (12 x^{2})^{2}

使用指数法则:

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

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

(x^{2})^{2} : x^{4}

使用指数法则:

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

= x^{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= x^{4}

= 1 2^{2} x^{4}

1 2^{2} = 144

= 144 x^{4}

2 \cdot 12 x^{2} \cdot 3 = 72 x^{2}

2 \cdot 12 x^{2} \cdot 3

数字相乘:

2 \cdot 12 \cdot 3 = 72

= 72 x^{2}

3^{2} = 9

3^{2}

3^{2} = 9

= 9

= 144 x^{4} + 72 x^{2} + 9

= 144 x^{4} + 72 x^{2} + 9

展开

(- 83 1 - x^{2} x)^{2} : 192 x^{2} - 192 x^{4}

(- 83 1 - x^{2} x)^{2}

使用指数法则:

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

若

n

是偶数

(- 83 1 - x^{2} x)^{2} = (83 1 - x^{2} x)^{2}

= (83 1 - x^{2} x)^{2}

使用指数法则:

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

= 8^{2} (3)^{2} x^{2} (1 - x^{2})^{2}

(3)^{2} : 3

使用根式运算法则:

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

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

使用指数法则:

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

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

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= 3

= 8^{2} \cdot 3 (1 - x^{2})^{2} x^{2}

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

使用根式运算法则:

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

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

使用指数法则:

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

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

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= 1 - x^{2}

= 8^{2} \cdot 3 (1 - x^{2}) x^{2}

整理后得

= 192 (1 - x^{2}) x^{2}

展开

192 (1 - x^{2}) x^{2} : 192 x^{2} - 192 x^{4}

192 (1 - x^{2}) x^{2}

= 192 x^{2} (1 - x^{2})

使用分配律:

a (b - c) = ab - a c

a = 192 x^{2}, b = 1, c = x^{2}

= 192 x^{2} \cdot 1 - 192 x^{2} x^{2}

= 192 \cdot 1 \cdot x^{2} - 192 x^{2} x^{2}

化简

192 \cdot 1 \cdot x^{2} - 192 x^{2} x^{2} : 192 x^{2} - 192 x^{4}

192 \cdot 1 \cdot x^{2} - 192 x^{2} x^{2}

192 \cdot 1 \cdot x^{2} = 192 x^{2}

192 \cdot 1 \cdot x^{2}

数字相乘:

192 \cdot 1 = 192

= 192 x^{2}

192 x^{2} x^{2} = 192 x^{4}

192 x^{2} x^{2}

使用指数法则:

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

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

= 192 x^{2 + 2}

数字相加:

2 + 2 = 4

= 192 x^{4}

= 192 x^{2} - 192 x^{4}

= 192 x^{2} - 192 x^{4}

= 192 x^{2} - 192 x^{4}

144 x^{4} + 72 x^{2} + 9 = 192 x^{2} - 192 x^{4}

144 x^{4} + 72 x^{2} + 9 = 192 x^{2} - 192 x^{4}

144 x^{4} + 72 x^{2} + 9 = 192 x^{2} - 192 x^{4}

解

144 x^{4} + 72 x^{2} + 9 = 192 x^{2} - 192 x^{4} : x = \frac{1}{2}, x = - \frac{1}{2}, x = \frac{3}{2 7}, x = - \frac{3}{2 7}

144 x^{4} + 72 x^{2} + 9 = 192 x^{2} - 192 x^{4}

将

192 x^{4}

para o lado esquerdo

144 x^{4} + 72 x^{2} + 9 = 192 x^{2} - 192 x^{4}

两边加上

192 x^{4}

144 x^{4} + 72 x^{2} + 9 + 192 x^{4} = 192 x^{2} - 192 x^{4} + 192 x^{4}

化简

336 x^{4} + 72 x^{2} + 9 = 192 x^{2}

336 x^{4} + 72 x^{2} + 9 = 192 x^{2}

将

192 x^{2}

para o lado esquerdo

336 x^{4} + 72 x^{2} + 9 = 192 x^{2}

两边减去

192 x^{2}

336 x^{4} + 72 x^{2} + 9 - 192 x^{2} = 192 x^{2} - 192 x^{2}

化简

336 x^{4} - 120 x^{2} + 9 = 0

336 x^{4} - 120 x^{2} + 9 = 0

用

u = x^{2}

和

u^{2} = x^{4}

改写方程式

336 u^{2} - 120 u + 9 = 0

解

336 u^{2} - 120 u + 9 = 0 : u = \frac{1}{4}, u = \frac{3}{28}

336 u^{2} - 120 u + 9 = 0

使用求根公式求解

336 u^{2} - 120 u + 9 = 0

二次方程求根公式:

若

a = 336, b = - 120, c = 9

u_{1, 2} = \frac{- ( - 120 ) \pm ( - 120 ) ^{2} - 4 \cdot 336 \cdot 9}{2 \cdot 336}

u_{1, 2} = \frac{- ( - 120 ) \pm ( - 120 ) ^{2} - 4 \cdot 336 \cdot 9}{2 \cdot 336}

(- 120)^{2} - 4 \cdot 336 \cdot 9 = 48

(- 120)^{2} - 4 \cdot 336 \cdot 9

使用指数法则:

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

若

n

是偶数

(- 120)^{2} = 12 0^{2}

= 12 0^{2} - 4 \cdot 336 \cdot 9

数字相乘:

4 \cdot 336 \cdot 9 = 12096

= 12 0^{2} - 12096

12 0^{2} = 14400

= 14400 - 12096

数字相减:

14400 - 12096 = 2304

= 2304

因式分解数字:

2304 = 4 8^{2}

= 4 8^{2}

使用根式运算法则:

4 8^{2} = 48

= 48

u_{1, 2} = \frac{- ( - 120 ) \pm 48}{2 \cdot 336}

将解分隔开

u_{1} = \frac{- ( - 120 ) + 48}{2 \cdot 336}, u_{2} = \frac{- ( - 120 ) - 48}{2 \cdot 336}

u = \frac{- ( - 120 ) + 48}{2 \cdot 336} : \frac{1}{4}

\frac{- ( - 120 ) + 48}{2 \cdot 336}

使用法则

- (- a) = a

= \frac{120 + 48}{2 \cdot 336}

数字相加:

120 + 48 = 168

= \frac{168}{2 \cdot 336}

数字相乘:

2 \cdot 336 = 672

= \frac{168}{672}

约分:

168

= \frac{1}{4}

u = \frac{- ( - 120 ) - 48}{2 \cdot 336} : \frac{3}{28}

\frac{- ( - 120 ) - 48}{2 \cdot 336}

使用法则

- (- a) = a

= \frac{120 - 48}{2 \cdot 336}

数字相减:

120 - 48 = 72

= \frac{72}{2 \cdot 336}

数字相乘:

2 \cdot 336 = 672

= \frac{72}{672}

约分:

24

= \frac{3}{28}

二次方程组的解是:

u = \frac{1}{4}, u = \frac{3}{28}

u = \frac{1}{4}, u = \frac{3}{28}

代回

u = x^{2}

，求解

x

解

x^{2} = \frac{1}{4} : x = \frac{1}{2}, x = - \frac{1}{2}

x^{2} = \frac{1}{4}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

x = \frac{1}{4}, x = - \frac{1}{4}

\frac{1}{4} = \frac{1}{2}

\frac{1}{4}

使用根式运算法则:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= \frac{1}{4}

使用根式运算法则:

1 = 1

1 = 1

= \frac{1}{4}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

a^{2} = a, a \geq 0

2^{2} = 2

= 2

= \frac{1}{2}

- \frac{1}{4} = - \frac{1}{2}

- \frac{1}{4}

使用根式运算法则:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= - \frac{1}{4}

使用根式运算法则:

1 = 1

1 = 1

= - \frac{1}{4}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

a^{2} = a, a \geq 0

2^{2} = 2

= 2

= - \frac{1}{2}

x = \frac{1}{2}, x = - \frac{1}{2}

解

x^{2} = \frac{3}{28} : x = \frac{3}{2 7}, x = - \frac{3}{2 7}

x^{2} = \frac{3}{28}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

x = \frac{3}{28}, x = - \frac{3}{28}

\frac{3}{28} = \frac{3}{2 7}

\frac{3}{28}

使用根式运算法则:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= \frac{3}{28}

28 = 27

28

28

质因数分解:

2^{2} \cdot 7

28

28

除以

228 = 14 \cdot 2

= 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 7

= 2^{2} \cdot 7

= 2^{2} \cdot 7

使用根式运算法则:

ab = a b, a \geq 0, b \geq 0

2^{2} \cdot 7 = 2^{2} 7

= 2^{2} 7

使用根式运算法则:

a^{2} = a, a \geq 0

2^{2} = 2

= 27

= \frac{3}{2 7}

- \frac{3}{28} = - \frac{3}{2 7}

- \frac{3}{28}

使用根式运算法则:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= - \frac{3}{28}

28 = 27

28

28

质因数分解:

2^{2} \cdot 7

28

28

除以

228 = 14 \cdot 2

= 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 7

= 2^{2} \cdot 7

= 2^{2} \cdot 7

使用根式运算法则:

ab = a b, a \geq 0, b \geq 0

2^{2} \cdot 7 = 2^{2} 7

= 2^{2} 7

使用根式运算法则:

a^{2} = a, a \geq 0

2^{2} = 2

= 27

= - \frac{3}{2 7}

x = \frac{3}{2 7}, x = - \frac{3}{2 7}

解为

x = \frac{1}{2}, x = - \frac{1}{2}, x = \frac{3}{2 7}, x = - \frac{3}{2 7}

x = \frac{1}{2}, x = - \frac{1}{2}, x = \frac{3}{2 7}, x = - \frac{3}{2 7}

验证解:

x = \frac{1}{2}

真

, x = - \frac{1}{2}

假

, x = \frac{3}{2 7}

真

, x = - \frac{3}{2 7}

假

将它们代入

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

检验解是否符合

去除与方程不符的解。

代入

x = \frac{1}{2} :

真

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

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

(\frac{1}{2}) 1 - (2 (\frac{1}{2}))^{2} + 2 (\frac{1}{2}) 1 - (\frac{1}{2})^{2}

去除括号:

(a) = a

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

\frac{1}{2} 1 - (2 \cdot \frac{1}{2})^{2} = 0

\frac{1}{2} 1 - (2 \cdot \frac{1}{2})^{2}

1 - (2 \cdot \frac{1}{2})^{2} = 0

1 - (2 \cdot \frac{1}{2})^{2}

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

(2 \cdot \frac{1}{2})^{2}

乘

2 \cdot \frac{1}{2} : 1

2 \cdot \frac{1}{2}

分式相乘:

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

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

约分:

2

= 1

= 1^{2}

使用法则

1^{a} = 1

= 1

= 1 - 1

数字相减:

1 - 1 = 0

= 0

使用法则

0 = 0

= 0

= 0 \cdot \frac{1}{2}

使用法则

0 \cdot a = 0

= 0

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

2 \cdot \frac{1}{2} 1 - (\frac{1}{2})^{2}

分式相乘:

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

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

约分:

2

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

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

1 - (\frac{1}{2})^{2}

(\frac{1}{2})^{2} = \frac{1}{4}

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

使用指数法则:

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

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

使用法则

1^{a} = 1

1^{2} = 1

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

2^{2} = 4

= \frac{1}{4}

= 1 - \frac{1}{4}

化简

1 - \frac{1}{4} : \frac{3}{4}

1 - \frac{1}{4}

将项转换为分式:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} - \frac{1}{4}

因为分母相等，所以合并分式:

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

= \frac{1 \cdot 4 - 1}{4}

1 \cdot 4 - 1 = 3

1 \cdot 4 - 1

数字相乘:

1 \cdot 4 = 4

= 4 - 1

数字相减:

4 - 1 = 3

= 3

= \frac{3}{4}

= \frac{3}{4}

使用根式运算法则:

假定

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

2^{2} = 2

= 2

= \frac{3}{2}

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

乘以:

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

= \frac{3}{2}

= 0 + \frac{3}{2}

0 + \frac{3}{2} = \frac{3}{2}

= \frac{3}{2}

\frac{3}{2} = \frac{3}{2}

真

代入

x = - \frac{1}{2} :

假

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

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

(- \frac{1}{2}) 1 - (2 (- \frac{1}{2}))^{2} + 2 (- \frac{1}{2}) 1 - (- \frac{1}{2})^{2}

去除括号:

(- a) = - a

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

\frac{1}{2} 1 - (- 2 \cdot \frac{1}{2})^{2} = 0

\frac{1}{2} 1 - (- 2 \cdot \frac{1}{2})^{2}

1 - (- 2 \cdot \frac{1}{2})^{2} = 0

1 - (- 2 \cdot \frac{1}{2})^{2}

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

(- 2 \cdot \frac{1}{2})^{2}

乘

- 2 \cdot \frac{1}{2} : - 1

- 2 \cdot \frac{1}{2}

分式相乘:

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

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

约分:

2

= - 1

= (- 1)^{2}

使用指数法则:

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

若

n

是偶数

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

= 1^{2}

使用法则

1^{a} = 1

= 1

= 1 - 1

数字相减:

1 - 1 = 0

= 0

使用法则

0 = 0

= 0

= 0 \cdot \frac{1}{2}

使用法则

0 \cdot a = 0

= 0

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

2 \cdot \frac{1}{2} 1 - (- \frac{1}{2})^{2}

分式相乘:

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

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

约分:

2

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

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

1 - (- \frac{1}{2})^{2}

(- \frac{1}{2})^{2} = \frac{1}{4}

(- \frac{1}{2})^{2}

使用指数法则:

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

若

n

是偶数

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

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

使用指数法则:

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

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

使用法则

1^{a} = 1

1^{2} = 1

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

2^{2} = 4

= \frac{1}{4}

= 1 - \frac{1}{4}

化简

1 - \frac{1}{4} : \frac{3}{4}

1 - \frac{1}{4}

将项转换为分式:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} - \frac{1}{4}

因为分母相等，所以合并分式:

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

= \frac{1 \cdot 4 - 1}{4}

1 \cdot 4 - 1 = 3

1 \cdot 4 - 1

数字相乘:

1 \cdot 4 = 4

= 4 - 1

数字相减:

4 - 1 = 3

= 3

= \frac{3}{4}

= \frac{3}{4}

使用根式运算法则:

假定

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

2^{2} = 2

= 2

= \frac{3}{2}

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

乘以:

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

= \frac{3}{2}

= - 0 - \frac{3}{2}

- 0 - \frac{3}{2} = - \frac{3}{2}

= - \frac{3}{2}

- \frac{3}{2} = \frac{3}{2}

假

代入

x = \frac{3}{2 7} :

真

(\frac{3}{2 7}) 1 - (2 (\frac{3}{2 7}))^{2} + 2 (\frac{3}{2 7}) 1 - (\frac{3}{2 7})^{2} = \frac{3}{2}

(\frac{3}{2 7}) 1 - (2 (\frac{3}{2 7}))^{2} + 2 (\frac{3}{2 7}) 1 - (\frac{3}{2 7})^{2} = \frac{3}{2}

(\frac{3}{2 7}) 1 - (2 (\frac{3}{2 7}))^{2} + 2 (\frac{3}{2 7}) 1 - (\frac{3}{2 7})^{2}

去除括号:

(a) = a

= \frac{3}{2 7} 1 - (2 \cdot \frac{3}{2 7})^{2} + 2 \cdot \frac{3}{2 7} 1 - (\frac{3}{2 7})^{2}

\frac{3}{2 7} 1 - (2 \cdot \frac{3}{2 7})^{2} = \frac{3}{7}

\frac{3}{2 7} 1 - (2 \cdot \frac{3}{2 7})^{2}

1 - (2 \cdot \frac{3}{2 7})^{2} = \frac{2}{7}

1 - (2 \cdot \frac{3}{2 7})^{2}

(2 \cdot \frac{3}{2 7})^{2} = \frac{3}{7}

(2 \cdot \frac{3}{2 7})^{2}

乘

2 \cdot \frac{3}{2 7} : \frac{3}{7}

2 \cdot \frac{3}{2 7}

分式相乘:

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

= \frac{3 \cdot 2}{2 7}

约分:

2

= \frac{3}{7}

合并相同指数项 :

\frac{x}{y} = \frac{x}{y}

= \frac{3}{7}

= (\frac{3}{7})^{2}

使用根式运算法则:

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

= ((\frac{3}{7})^{\frac{1}{2}})^{2}

使用指数法则:

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

= (\frac{3}{7})^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= \frac{3}{7}

= 1 - \frac{3}{7}

化简

1 - \frac{3}{7} : \frac{4}{7}

1 - \frac{3}{7}

将项转换为分式:

1 = \frac{1 \cdot 7}{7}

= \frac{1 \cdot 7}{7} - \frac{3}{7}

因为分母相等，所以合并分式:

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

= \frac{1 \cdot 7 - 3}{7}

1 \cdot 7 - 3 = 4

1 \cdot 7 - 3

数字相乘:

1 \cdot 7 = 7

= 7 - 3

数字相减:

7 - 3 = 4

= 4

= \frac{4}{7}

= \frac{4}{7}

使用根式运算法则:

假定

a \geq 0, b \geq 0

= \frac{4}{7}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

2^{2} = 2

= 2

= \frac{2}{7}

= \frac{2}{7} \cdot \frac{3}{2 7}

分式相乘:

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

= \frac{3 \cdot 2}{2 7 7}

约分:

2

= \frac{3}{7 7}

77 = 7

77

使用根式运算法则:

a a = a

77 = 7

= 7

= \frac{3}{7}

2 \cdot \frac{3}{2 7} 1 - (\frac{3}{2 7})^{2} = \frac{5 3}{14}

2 \cdot \frac{3}{2 7} 1 - (\frac{3}{2 7})^{2}

分式相乘:

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

= \frac{3 \cdot 2 1 - ( \frac{3}{2 7} ) ^{2}}{2 7}

约分:

2

= \frac{3 1 - ( \frac{3}{2 7} ) ^{2}}{7}

1 - (\frac{3}{2 7})^{2} = \frac{5}{2 7}

1 - (\frac{3}{2 7})^{2}

(\frac{3}{2 7})^{2} = \frac{3}{28}

(\frac{3}{2 7})^{2}

使用指数法则:

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

= \frac{( 3 ) ^{2}}{( 2 7 ) ^{2}}

使用指数法则:

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

(27)^{2} = 2^{2} (7)^{2}

= \frac{( 3 ) ^{2}}{2 ^{2} ( 7 ) ^{2}}

(3)^{2} : 3

使用根式运算法则:

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

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

使用指数法则:

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

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

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= 3

= \frac{3}{2 ^{2} ( 7 ) ^{2}}

(7)^{2} : 7

使用根式运算法则:

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

= (7^{\frac{1}{2}})^{2}

使用指数法则:

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

= 7^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= 7

= \frac{3}{2 ^{2} \cdot 7}

2^{2} \cdot 7 = 28

2^{2} \cdot 7

2^{2} = 4

= 4 \cdot 7

数字相乘:

4 \cdot 7 = 28

= 28

= \frac{3}{28}

= 1 - \frac{3}{28}

化简

1 - \frac{3}{28} : \frac{25}{28}

1 - \frac{3}{28}

将项转换为分式:

1 = \frac{1 \cdot 28}{28}

= \frac{1 \cdot 28}{28} - \frac{3}{28}

因为分母相等，所以合并分式:

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

= \frac{1 \cdot 28 - 3}{28}

1 \cdot 28 - 3 = 25

1 \cdot 28 - 3

数字相乘:

1 \cdot 28 = 28

= 28 - 3

数字相减:

28 - 3 = 25

= 25

= \frac{25}{28}

= \frac{25}{28}

使用根式运算法则:

假定

a \geq 0, b \geq 0

= \frac{25}{28}

28 = 27

28

28

质因数分解:

2^{2} \cdot 7

28

28

除以

228 = 14 \cdot 2

= 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 7

= 2^{2} \cdot 7

= 2^{2} \cdot 7

使用根式运算法则:

= 7 2^{2}

使用根式运算法则:

2^{2} = 2

= 27

= \frac{25}{2 7}

25 = 5

25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

5^{2} = 5

= 5

= \frac{5}{2 7}

= \frac{3 \frac{5}{2 7}}{7}

乘

3 \frac{5}{2 7} : \frac{5 3}{2 7}

3 \frac{5}{2 7}

分式相乘:

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

= \frac{5 3}{2 7}

= \frac{\frac{5 3}{2 7}}{7}

使用分式法则:

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

= \frac{5 3}{2 7 7}

277 = 14

277

使用根式运算法则:

a a = a

77 = 7

= 2 \cdot 7

数字相乘:

2 \cdot 7 = 14

= 14

= \frac{5 3}{14}

= \frac{3}{7} + \frac{5 3}{14}

7, 14

的最小公倍数:

14

7, 14

最小公倍数 (LCM)

7

质因数分解:

7

7

7

是质数，因此无法因数分解

= 7

14

质因数分解:

2 \cdot 7

14

14

除以

214 = 7 \cdot 2

= 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 7

将每个因子乘以它在

7

或

14

中出现的最多次数

= 7 \cdot 2

数字相乘:

7 \cdot 2 = 14

= 14

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

14

对于

\frac{3}{7} :

将分母和分子乘以

2

\frac{3}{7} = \frac{3 \cdot 2}{7 \cdot 2} = \frac{3 \cdot 2}{14}

= \frac{3 \cdot 2}{14} + \frac{5 3}{14}

因为分母相等，所以合并分式:

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

= \frac{3 \cdot 2 + 5 3}{14}

同类项相加:

23 + 53 = 73

= \frac{7 3}{14}

约分:

7

= \frac{3}{2}

\frac{3}{2} = \frac{3}{2}

真

代入

x = - \frac{3}{2 7} :

假

(- \frac{3}{2 7}) 1 - (2 (- \frac{3}{2 7}))^{2} + 2 (- \frac{3}{2 7}) 1 - (- \frac{3}{2 7})^{2} = \frac{3}{2}

(- \frac{3}{2 7}) 1 - (2 (- \frac{3}{2 7}))^{2} + 2 (- \frac{3}{2 7}) 1 - (- \frac{3}{2 7})^{2} = - \frac{3}{2}

(- \frac{3}{2 7}) 1 - (2 (- \frac{3}{2 7}))^{2} + 2 (- \frac{3}{2 7}) 1 - (- \frac{3}{2 7})^{2}

去除括号:

(- a) = - a

= - \frac{3}{2 7} 1 - (- 2 \cdot \frac{3}{2 7})^{2} - 2 \cdot \frac{3}{2 7} 1 - (- \frac{3}{2 7})^{2}

\frac{3}{2 7} 1 - (- 2 \cdot \frac{3}{2 7})^{2} = \frac{3}{7}

\frac{3}{2 7} 1 - (- 2 \cdot \frac{3}{2 7})^{2}

1 - (- 2 \cdot \frac{3}{2 7})^{2} = \frac{2}{7}

1 - (- 2 \cdot \frac{3}{2 7})^{2}

(- 2 \cdot \frac{3}{2 7})^{2} = \frac{3}{7}

(- 2 \cdot \frac{3}{2 7})^{2}

乘

- 2 \cdot \frac{3}{2 7} : - \frac{3}{7}

- 2 \cdot \frac{3}{2 7}

分式相乘:

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

= - \frac{3 \cdot 2}{2 7}

约分:

2

= - \frac{3}{7}

合并相同指数项 :

\frac{x}{y} = \frac{x}{y}

= - \frac{3}{7}

= (- \frac{3}{7})^{2}

使用指数法则:

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

若

n

是偶数

(- \frac{3}{7})^{2} = (\frac{3}{7})^{2}

= (\frac{3}{7})^{2}

使用根式运算法则:

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

= ((\frac{3}{7})^{\frac{1}{2}})^{2}

使用指数法则:

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

= (\frac{3}{7})^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= \frac{3}{7}

= 1 - \frac{3}{7}

化简

1 - \frac{3}{7} : \frac{4}{7}

1 - \frac{3}{7}

将项转换为分式:

1 = \frac{1 \cdot 7}{7}

= \frac{1 \cdot 7}{7} - \frac{3}{7}

因为分母相等，所以合并分式:

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

= \frac{1 \cdot 7 - 3}{7}

1 \cdot 7 - 3 = 4

1 \cdot 7 - 3

数字相乘:

1 \cdot 7 = 7

= 7 - 3

数字相减:

7 - 3 = 4

= 4

= \frac{4}{7}

= \frac{4}{7}

使用根式运算法则:

假定

a \geq 0, b \geq 0

= \frac{4}{7}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

2^{2} = 2

= 2

= \frac{2}{7}

= \frac{2}{7} \cdot \frac{3}{2 7}

分式相乘:

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

= \frac{3 \cdot 2}{2 7 7}

约分:

2

= \frac{3}{7 7}

77 = 7

77

使用根式运算法则:

a a = a

77 = 7

= 7

= \frac{3}{7}

2 \cdot \frac{3}{2 7} 1 - (- \frac{3}{2 7})^{2} = \frac{5 3}{14}

2 \cdot \frac{3}{2 7} 1 - (- \frac{3}{2 7})^{2}

分式相乘:

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

= \frac{3 \cdot 2 1 - ( - \frac{3}{2 7} ) ^{2}}{2 7}

约分:

2

= \frac{3 1 - ( - \frac{3}{2 7} ) ^{2}}{7}

3 1 - (- \frac{3}{2 7})^{2} = 3 1 - (\frac{3}{2 7})^{2}

3 1 - (- \frac{3}{2 7})^{2}

1 - (- \frac{3}{2 7})^{2} = 1 - (\frac{3}{2 7})^{2}

1 - (- \frac{3}{2 7})^{2}

使用指数法则:

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

若

n

是偶数

(- \frac{3}{2 7})^{2} = (\frac{3}{2 7})^{2}

= 1 - (\frac{3}{2 7})^{2}

= 3 - (\frac{3}{2 7})^{2} + 1

= \frac{3 - ( \frac{3}{2 7} ) ^{2} + 1}{7}

1 - (\frac{3}{2 7})^{2} = \frac{5}{2 7}

1 - (\frac{3}{2 7})^{2}

(\frac{3}{2 7})^{2} = \frac{3}{28}

(\frac{3}{2 7})^{2}

使用指数法则:

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

= \frac{( 3 ) ^{2}}{( 2 7 ) ^{2}}

使用指数法则:

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

(27)^{2} = 2^{2} (7)^{2}

= \frac{( 3 ) ^{2}}{2 ^{2} ( 7 ) ^{2}}

(3)^{2} : 3

使用根式运算法则:

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

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

使用指数法则:

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

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

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= 3

= \frac{3}{2 ^{2} ( 7 ) ^{2}}

(7)^{2} : 7

使用根式运算法则:

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

= (7^{\frac{1}{2}})^{2}

使用指数法则:

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

= 7^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= 7

= \frac{3}{2 ^{2} \cdot 7}

2^{2} \cdot 7 = 28

2^{2} \cdot 7

2^{2} = 4

= 4 \cdot 7

数字相乘:

4 \cdot 7 = 28

= 28

= \frac{3}{28}

= 1 - \frac{3}{28}

化简

1 - \frac{3}{28} : \frac{25}{28}

1 - \frac{3}{28}

将项转换为分式:

1 = \frac{1 \cdot 28}{28}

= \frac{1 \cdot 28}{28} - \frac{3}{28}

因为分母相等，所以合并分式:

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

= \frac{1 \cdot 28 - 3}{28}

1 \cdot 28 - 3 = 25

1 \cdot 28 - 3

数字相乘:

1 \cdot 28 = 28

= 28 - 3

数字相减:

28 - 3 = 25

= 25

= \frac{25}{28}

= \frac{25}{28}

使用根式运算法则:

假定

a \geq 0, b \geq 0

= \frac{25}{28}

28 = 27

28

28

质因数分解:

2^{2} \cdot 7

28

28

除以

228 = 14 \cdot 2

= 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 7

= 2^{2} \cdot 7

= 2^{2} \cdot 7

使用根式运算法则:

= 7 2^{2}

使用根式运算法则:

2^{2} = 2

= 27

= \frac{25}{2 7}

25 = 5

25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

5^{2} = 5

= 5

= \frac{5}{2 7}

= \frac{3 \frac{5}{2 7}}{7}

乘

3 \frac{5}{2 7} : \frac{5 3}{2 7}

3 \frac{5}{2 7}

分式相乘:

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

= \frac{5 3}{2 7}

= \frac{\frac{5 3}{2 7}}{7}

使用分式法则:

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

= \frac{5 3}{2 7 7}

277 = 14

277

使用根式运算法则:

a a = a

77 = 7

= 2 \cdot 7

数字相乘:

2 \cdot 7 = 14

= 14

= \frac{5 3}{14}

= - \frac{3}{7} - \frac{5 3}{14}

7, 14

的最小公倍数:

14

7, 14

最小公倍数 (LCM)

7

质因数分解:

7

7

7

是质数，因此无法因数分解

= 7

14

质因数分解:

2 \cdot 7

14

14

除以

214 = 7 \cdot 2

= 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 7

将每个因子乘以它在

7

或

14

中出现的最多次数

= 7 \cdot 2

数字相乘:

7 \cdot 2 = 14

= 14

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

14

对于

\frac{3}{7} :

将分母和分子乘以

2

\frac{3}{7} = \frac{3 \cdot 2}{7 \cdot 2} = \frac{3 \cdot 2}{14}

= - \frac{3 \cdot 2}{14} - \frac{5 3}{14}

因为分母相等，所以合并分式:

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

= \frac{- 3 \cdot 2 - 5 3}{14}

同类项相加:

- 23 - 53 = - 73

= \frac{- 7 3}{14}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{7 3}{14}

约分:

7

= - \frac{3}{2}

- \frac{3}{2} = \frac{3}{2}

假

解为

x = \frac{1}{2}, x = \frac{3}{2 7}

x = \frac{1}{2}, x = \frac{3}{2 7}

将解代入原方程进行验证

将它们代入

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

检验解是否符合

去除与方程不符的解。

检验

\frac{1}{2}

的解:

假

\frac{1}{2}

代入

n = 1

\frac{1}{2}

对于

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

代入

x = \frac{1}{2}

arcsin (\frac{1}{2}) + arcsin (2 \cdot \frac{1}{2}) = \frac{π}{3}

整理后得

2.09439 \dots = 1.04719 \dots

\Rightarrow 假

检验

\frac{3}{2 7}

的解:

真

\frac{3}{2 7}

代入

n = 1

\frac{3}{2 7}

对于

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

代入

x = \frac{3}{2 7}

arcsin (\frac{3}{2 7}) + arcsin (2 \cdot \frac{3}{2 7}) = \frac{π}{3}

整理后得

1.04719 \dots = 1.04719 \dots

\Rightarrow 真

x = \frac{3}{2 7}

作图

流行的例子

cos(x)-sqrt(1-3cos^2(x))=0

arctan(2x-3)= pi/4

tan(2x)+2cos(x)=0,0<= x<= 2pi