What is the general solution for sec^2(x)+0.5sec(x)=0 ?

The general solution for sec^2(x)+0.5sec(x)=0 is No Solution for x\in\mathbb{R}

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Popular Trigonometry >

sec^2(x)+0.5sec(x)=0

Solution

sec^{2} (x) + 0.5 sec (x) = 0

Solution

No Solution for x \in R

Solution steps

sec^{2} (x) + 0.5 sec (x) = 0

Solve by substitution

sec^{2} (x) + 0.5 sec (x) = 0

Let:

sec (x) = u

u^{2} + 0.5 u = 0

u^{2} + 0.5 u = 0 : u = 0, u = - \frac{1}{2}

u^{2} + 0.5 u = 0

Multiply both sides by

10

u^{2} + 0.5 u = 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

u^{2} \cdot 10 + 0.5 u \cdot 10 = 0 \cdot 10

Refine

10 u^{2} + 5 u = 0

10 u^{2} + 5 u = 0

Solve with the quadratic formula

10 u^{2} + 5 u = 0

Quadratic Equation Formula:

For

a = 10, b = 5, c = 0

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 10 \cdot 0}{2 \cdot 10}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 10 \cdot 0}{2 \cdot 10}

5^{2} - 4 \cdot 10 \cdot 0 = 5

5^{2} - 4 \cdot 10 \cdot 0

Apply rule

0 \cdot a = 0

= 5^{2} - 0

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

= 5^{2}

Apply radical rule:

assuming

a \geq 0

= 5

u_{1, 2} = \frac{- 5 \pm 5}{2 \cdot 10}

Separate the solutions

u_{1} = \frac{- 5 + 5}{2 \cdot 10}, u_{2} = \frac{- 5 - 5}{2 \cdot 10}

u = \frac{- 5 + 5}{2 \cdot 10} : 0

\frac{- 5 + 5}{2 \cdot 10}

Add/Subtract the numbers:

- 5 + 5 = 0

= \frac{0}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{0}{20}

Apply rule

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

= 0

u = \frac{- 5 - 5}{2 \cdot 10} : - \frac{1}{2}

\frac{- 5 - 5}{2 \cdot 10}

Subtract the numbers:

- 5 - 5 = - 10

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

Multiply the numbers:

2 \cdot 10 = 20

= \frac{- 10}{20}

Apply the fraction rule:

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

= - \frac{10}{20}

Cancel the common factor:

10

= - \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

Graph

Popular Examples

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Frequently Asked Questions (FAQ)

What is the general solution for sec^2(x)+0.5sec(x)=0 ?
The general solution for sec^2(x)+0.5sec(x)=0 is No Solution for x\in\mathbb{R}