What is the general solution for 2sin^2(x)-5sin(x)-12=0 ?

The general solution for 2sin^2(x)-5sin(x)-12=0 is No Solution for x\in\mathbb{R}

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

2sin^2(x)-5sin(x)-12=0

Solution

2 sin^{2} (x) - 5 sin (x) - 12 = 0

Solution

No Solution for x \in R

Solution steps

2 sin^{2} (x) - 5 sin (x) - 12 = 0

Solve by substitution

2 sin^{2} (x) - 5 sin (x) - 12 = 0

Let:

sin (x) = u

2 u^{2} - 5 u - 12 = 0

2 u^{2} - 5 u - 12 = 0 : u = 4, u = - \frac{3}{2}

2 u^{2} - 5 u - 12 = 0

Solve with the quadratic formula

2 u^{2} - 5 u - 12 = 0

Quadratic Equation Formula:

For

a = 2, b = - 5, c = - 12

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 2 ( - 12 )}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 2 ( - 12 )}{2 \cdot 2}

(- 5)^{2} - 4 \cdot 2 (- 12) = 11

(- 5)^{2} - 4 \cdot 2 (- 12)

Apply rule

- (- a) = a

= (- 5)^{2} + 4 \cdot 2 \cdot 12

Apply exponent rule:

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

n

is even

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 2 \cdot 12

Multiply the numbers:

4 \cdot 2 \cdot 12 = 96

= 5^{2} + 96

5^{2} = 25

= 25 + 96

Add the numbers:

25 + 96 = 121

= 121

Factor the number:

121 = 1 1^{2}

= 1 1^{2}

Apply radical rule:

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- ( - 5 ) \pm 11}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- ( - 5 ) + 11}{2 \cdot 2}, u_{2} = \frac{- ( - 5 ) - 11}{2 \cdot 2}

u = \frac{- ( - 5 ) + 11}{2 \cdot 2} : 4

\frac{- ( - 5 ) + 11}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{5 + 11}{2 \cdot 2}

Add the numbers:

5 + 11 = 16

= \frac{16}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{16}{4}

Divide the numbers:

\frac{16}{4} = 4

= 4

u = \frac{- ( - 5 ) - 11}{2 \cdot 2} : - \frac{3}{2}

\frac{- ( - 5 ) - 11}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{5 - 11}{2 \cdot 2}

Subtract the numbers:

5 - 11 = - 6

= \frac{- 6}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 6}{4}

Apply the fraction rule:

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

= - \frac{6}{4}

Cancel the common factor:

2

= - \frac{3}{2}

The solutions to the quadratic equation are:

u = 4, u = - \frac{3}{2}

Substitute back

u = sin (x)

sin (x) = 4, sin (x) = - \frac{3}{2}

sin (x) = 4, sin (x) = - \frac{3}{2}

sin (x) = 4 :

No Solution

sin (x) = 4

- 1 \leq sin (x) \leq 1

No Solution

sin (x) = - \frac{3}{2} :

No Solution

sin (x) = - \frac{3}{2}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

No Solution for x \in R

Popular Examples

1/(sin(x))=sqrt(5)

\frac{1}{sin ( x )} = 5

tan(2x)+tan(4x)=0

tan (2 x) + tan (4 x) = 0

2sin(θ)-sqrt(3)=0,0<= θ<2pi

2 sin (θ) - 3 = 0, 0 \leq θ < 2 π

arctan(θ)=-1

arctan (θ) = - 1

tan^2(θ)-tan(θ)-2=0

tan^{2} (θ) - tan (θ) - 2 = 0

Frequently Asked Questions (FAQ)

What is the general solution for 2sin^2(x)-5sin(x)-12=0 ?
The general solution for 2sin^2(x)-5sin(x)-12=0 is No Solution for x\in\mathbb{R}