What is the general solution for sec^2(x)+1/2*sec(x)=0 ?

The general solution for sec^2(x)+1/2*sec(x)=0 is No Solution for x\in\mathbb{R}

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sec^2(x)+1/2*sec(x)=0

Solution

sec^{2} (x) + \frac{1}{2} \cdot sec (x) = 0

Solution

No Solution for x \in R

Solution steps

sec^{2} (x) + \frac{1}{2} sec (x) = 0

Solve by substitution

sec^{2} (x) + \frac{1}{2} sec (x) = 0

Let:

sec (x) = u

u^{2} + \frac{1}{2} u = 0

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

u^{2} + \frac{1}{2} u = 0

Multiply both sides by

2

u^{2} + \frac{1}{2} u = 0

Multiply both sides by

2

u^{2} \cdot 2 + \frac{1}{2} u \cdot 2 = 0 \cdot 2

Simplify

2 u^{2} + u = 0

2 u^{2} + u = 0

Solve with the quadratic formula

2 u^{2} + u = 0

Quadratic Equation Formula:

For

a = 2, b = 1, c = 0

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

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

1^{2} - 4 \cdot 2 \cdot 0 = 1

1^{2} - 4 \cdot 2 \cdot 0

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 2 \cdot 0

Apply rule

0 \cdot a = 0

= 1 - 0

Subtract the numbers:

1 - 0 = 1

= 1

Apply rule

1 = 1

= 1

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

Separate the solutions

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

u = \frac{- 1 + 1}{2 \cdot 2} : 0

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

Add/Subtract the numbers:

- 1 + 1 = 0

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{0}{4}

Apply rule

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

= 0

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

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

Subtract the numbers:

- 1 - 1 = - 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 2}{4}

Apply the fraction rule:

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

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

The solutions to the quadratic equation are:

u = 0, u = - \frac{1}{2}

Substitute back

u = sec (x)

sec (x) = 0, sec (x) = - \frac{1}{2}

sec (x) = 0, sec (x) = - \frac{1}{2}

sec (x) = 0 :

No Solution

sec (x) = 0

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

No Solution

sec (x) = - \frac{1}{2} :

No Solution

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

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

No Solution

Combine all the solutions

No Solution for x \in R

Popular Examples

2sin^2(x)-9sin(x)-5=0

2 sin^{2} (x) - 9 sin (x) - 5 = 0

2sin(a)-sqrt(3)=0

2 sin (a) - 3 = 0

sin(x)=-sqrt(3/2)

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

-pi/2 =picos(θ)

- \frac{π}{2} = π cos (θ)

2sin^2(x)=5cos(x)-1

2 sin^{2} (x) = 5 cos (x) - 1

Frequently Asked Questions (FAQ)

What is the general solution for sec^2(x)+1/2*sec(x)=0 ?
The general solution for sec^2(x)+1/2*sec(x)=0 is No Solution for x\in\mathbb{R}