What is the general solution for (4sec^2(x))/2 =-8sec(x) ?

The general solution for (4sec^2(x))/2 =-8sec(x) is x=1.82347…+2pin,x=-1.82347…+2pin

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(4sec^2(x))/2 =-8sec(x)

Solution

\frac{4 sec ^{2} ( x )}{2} = - 8 sec (x)

Solution

x = 1.82347 \dots + 2 πn, x = - 1.82347 \dots + 2 πn

Degrees

x = 104.47751 \dots^{\circ} + 36 0^{\circ} n, x = - 104.47751 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{4 sec ^{2} ( x )}{2} = - 8 sec (x)

Solve by substitution

\frac{4 sec ^{2} ( x )}{2} = - 8 sec (x)

Let:

sec (x) = u

\frac{4 u ^{2}}{2} = - 8 u

\frac{4 u ^{2}}{2} = - 8 u : u = 0, u = - 4

\frac{4 u ^{2}}{2} = - 8 u

Multiply both sides by

2

\frac{4 u ^{2}}{2} = - 8 u

Multiply both sides by

2

\frac{4 u ^{2}}{2} \cdot 2 = - 8 u \cdot 2

Simplify

4 u^{2} = - 16 u

4 u^{2} = - 16 u

Move

16 u

to the left side

4 u^{2} = - 16 u

Add

16 u

to both sides

4 u^{2} + 16 u = - 16 u + 16 u

Simplify

4 u^{2} + 16 u = 0

4 u^{2} + 16 u = 0

Solve with the quadratic formula

4 u^{2} + 16 u = 0

Quadratic Equation Formula:

For

a = 4, b = 16, c = 0

u_{1, 2} = \frac{- 16 \pm 1 6 ^{2} - 4 \cdot 4 \cdot 0}{2 \cdot 4}

u_{1, 2} = \frac{- 16 \pm 1 6 ^{2} - 4 \cdot 4 \cdot 0}{2 \cdot 4}

1 6^{2} - 4 \cdot 4 \cdot 0 = 16

1 6^{2} - 4 \cdot 4 \cdot 0

Apply rule

0 \cdot a = 0

= 1 6^{2} - 0

1 6^{2} - 0 = 1 6^{2}

= 1 6^{2}

Apply radical rule:

assuming

a \geq 0

= 16

u_{1, 2} = \frac{- 16 \pm 16}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- 16 + 16}{2 \cdot 4}, u_{2} = \frac{- 16 - 16}{2 \cdot 4}

u = \frac{- 16 + 16}{2 \cdot 4} : 0

\frac{- 16 + 16}{2 \cdot 4}

Add/Subtract the numbers:

- 16 + 16 = 0

= \frac{0}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{0}{8}

Apply rule

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

= 0

u = \frac{- 16 - 16}{2 \cdot 4} : - 4

\frac{- 16 - 16}{2 \cdot 4}

Subtract the numbers:

- 16 - 16 = - 32

= \frac{- 32}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 32}{8}

Apply the fraction rule:

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

= - \frac{32}{8}

Divide the numbers:

\frac{32}{8} = 4

= - 4

The solutions to the quadratic equation are:

u = 0, u = - 4

Substitute back

u = sec (x)

sec (x) = 0, sec (x) = - 4

sec (x) = 0, sec (x) = - 4

sec (x) = 0 :

No Solution

sec (x) = 0

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

No Solution

sec (x) = - 4 : x = arcsec (- 4) + 2 πn, x = - arcsec (- 4) + 2 πn

sec (x) = - 4

Apply trig inverse properties

sec (x) = - 4

General solutions for

sec (x) = - 4

sec (x) = - a \Rightarrow x = arcsec (- a) + 2 πn, x = - arcsec (- a) + 2 πn

x = arcsec (- 4) + 2 πn, x = - arcsec (- 4) + 2 πn

x = arcsec (- 4) + 2 πn, x = - arcsec (- 4) + 2 πn

Combine all the solutions

x = arcsec (- 4) + 2 πn, x = - arcsec (- 4) + 2 πn

Show solutions in decimal form

x = 1.82347 \dots + 2 πn, x = - 1.82347 \dots + 2 πn

Popular Examples

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tan (x) + tan^{3} (x) = 0

sin^3(x)-2sin(x)=4cos^2(x)-3

sin^{3} (x) - 2 sin (x) = 4 cos^{2} (x) - 3

cos^2(x)+6sin(x)-6=0

cos^{2} (x) + 6 sin (x) - 6 = 0

(1-sin(x))*cos^3(x)=0

(1 - sin (x)) \cdot cos^{3} (x) = 0

-tan(x)=3.465

- tan (x) = 3.465

Frequently Asked Questions (FAQ)

What is the general solution for (4sec^2(x))/2 =-8sec(x) ?
The general solution for (4sec^2(x))/2 =-8sec(x) is x=1.82347…+2pin,x=-1.82347…+2pin