What is the general solution for 3sec^2(x)=sec(x) ?

The general solution for 3sec^2(x)=sec(x) is No Solution for x\in\mathbb{R}

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Popular Trigonometry >

3sec^2(x)=sec(x)

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Solve by substitution

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

Let:

sec (x) = u

3 u^{2} = u

3 u^{2} = u : u = \frac{1}{3}, u = 0

3 u^{2} = u

Move

u

to the left side

3 u^{2} = u

Subtract

u

from both sides

3 u^{2} - u = u - u

Simplify

3 u^{2} - u = 0

3 u^{2} - u = 0

Solve with the quadratic formula

3 u^{2} - u = 0

Quadratic Equation Formula:

For

a = 3, b = - 1, c = 0

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 3 \cdot 0}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 3 \cdot 0}{2 \cdot 3}

(- 1)^{2} - 4 \cdot 3 \cdot 0 = 1

(- 1)^{2} - 4 \cdot 3 \cdot 0

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

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

n

is even

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

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 3 \cdot 0 = 0

4 \cdot 3 \cdot 0

Apply rule

0 \cdot a = 0

= 0

= 1 - 0

Subtract the numbers:

1 - 0 = 1

= 1

Apply rule

1 = 1

= 1

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

Separate the solutions

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

u = \frac{- ( - 1 ) + 1}{2 \cdot 3} : \frac{1}{3}

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

Apply rule

- (- a) = a

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

Add the numbers:

1 + 1 = 2

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{2}{6}

Cancel the common factor:

2

= \frac{1}{3}

u = \frac{- ( - 1 ) - 1}{2 \cdot 3} : 0

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

Apply rule

- (- a) = a

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

Subtract the numbers:

1 - 1 = 0

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{0}{6}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

The solutions to the quadratic equation are:

u = \frac{1}{3}, u = 0

Substitute back

u = sec (x)

sec (x) = \frac{1}{3}, sec (x) = 0

sec (x) = \frac{1}{3}, sec (x) = 0

sec (x) = \frac{1}{3} :

No Solution

sec (x) = \frac{1}{3}

sec (x) \leq - 1 or sec (x) \geq 1

No Solution

sec (x) = 0 :

No Solution

sec (x) = 0

sec (x) \leq - 1 or sec (x) \geq 1

No Solution

Combine all the solutions

No Solution for x \in R

Graph

Popular Examples

3cos^2(x)-8cos(x)+4=0,0<= x<= 180

Frequently Asked Questions (FAQ)

What is the general solution for 3sec^2(x)=sec(x) ?
The general solution for 3sec^2(x)=sec(x) is No Solution for x\in\mathbb{R}