What is the general solution for 3sin^2(x)+11sin(x)+10=0 ?

The general solution for 3sin^2(x)+11sin(x)+10=0 is No Solution for x\in\mathbb{R}

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Popular >

3sin^2(x)+11sin(x)+10=0

Solution

3 sin^{2} (x) + 11 sin (x) + 10 = 0

Solution

No Solution for x \in R

Solution steps

3 sin^{2} (x) + 11 sin (x) + 10 = 0

Solve by substitution

3 sin^{2} (x) + 11 sin (x) + 10 = 0

Let:

sin (x) = u

3 u^{2} + 11 u + 10 = 0

3 u^{2} + 11 u + 10 = 0 : u = - \frac{5}{3}, u = - 2

3 u^{2} + 11 u + 10 = 0

Solve with the quadratic formula

3 u^{2} + 11 u + 10 = 0

Quadratic Equation Formula:

For

a = 3, b = 11, c = 10

u_{1, 2} = \frac{- 11 \pm 1 1 ^{2} - 4 \cdot 3 \cdot 10}{2 \cdot 3}

u_{1, 2} = \frac{- 11 \pm 1 1 ^{2} - 4 \cdot 3 \cdot 10}{2 \cdot 3}

1 1^{2} - 4 \cdot 3 \cdot 10 = 1

1 1^{2} - 4 \cdot 3 \cdot 10

Multiply the numbers:

4 \cdot 3 \cdot 10 = 120

= 1 1^{2} - 120

1 1^{2} = 121

= 121 - 120

Subtract the numbers:

121 - 120 = 1

= 1

Apply rule

1 = 1

= 1

u_{1, 2} = \frac{- 11 \pm 1}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- 11 + 1}{2 \cdot 3}, u_{2} = \frac{- 11 - 1}{2 \cdot 3}

u = \frac{- 11 + 1}{2 \cdot 3} : - \frac{5}{3}

\frac{- 11 + 1}{2 \cdot 3}

Add/Subtract the numbers:

- 11 + 1 = - 10

= \frac{- 10}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 10}{6}

Apply the fraction rule:

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

= - \frac{10}{6}

Cancel the common factor:

2

= - \frac{5}{3}

u = \frac{- 11 - 1}{2 \cdot 3} : - 2

\frac{- 11 - 1}{2 \cdot 3}

Subtract the numbers:

- 11 - 1 = - 12

= \frac{- 12}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 12}{6}

Apply the fraction rule:

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

= - \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= - 2

The solutions to the quadratic equation are:

u = - \frac{5}{3}, u = - 2

Substitute back

u = sin (x)

sin (x) = - \frac{5}{3}, sin (x) = - 2

sin (x) = - \frac{5}{3}, sin (x) = - 2

sin (x) = - \frac{5}{3} :

No Solution

sin (x) = - \frac{5}{3}

- 1 \leq sin (x) \leq 1

No Solution

sin (x) = - 2 :

No Solution

sin (x) = - 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for 3sin^2(x)+11sin(x)+10=0 ?
The general solution for 3sin^2(x)+11sin(x)+10=0 is No Solution for x\in\mathbb{R}