What is the general solution for -5sec^2(x)-10sec(x)=-15 ?

The general solution for -5sec^2(x)-10sec(x)=-15 is x=1.91063…+2pin,x=-1.91063…+2pin,x=2pin

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Popular >

-5sec^2(x)-10sec(x)=-15

Solution

- 5 sec^{2} (x) - 10 sec (x) = - 15

Solution

x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn, x = 2 πn

Degrees

x = 109.47122 \dots^{\circ} + 36 0^{\circ} n, x = - 109.47122 \dots^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n

Solution steps

- 5 sec^{2} (x) - 10 sec (x) = - 15

Solve by substitution

- 5 sec^{2} (x) - 10 sec (x) = - 15

Let:

sec (x) = u

- 5 u^{2} - 10 u = - 15

- 5 u^{2} - 10 u = - 15 : u = - 3, u = 1

- 5 u^{2} - 10 u = - 15

Move

15

to the left side

- 5 u^{2} - 10 u = - 15

Add

15

to both sides

- 5 u^{2} - 10 u + 15 = - 15 + 15

Simplify

- 5 u^{2} - 10 u + 15 = 0

- 5 u^{2} - 10 u + 15 = 0

Solve with the quadratic formula

- 5 u^{2} - 10 u + 15 = 0

Quadratic Equation Formula:

For

a = - 5, b = - 10, c = 15

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 ( - 5 ) \cdot 15}{2 ( - 5 )}

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 ( - 5 ) \cdot 15}{2 ( - 5 )}

(- 10)^{2} - 4 (- 5) \cdot 15 = 20

(- 10)^{2} - 4 (- 5) \cdot 15

Apply rule

- (- a) = a

= (- 10)^{2} + 4 \cdot 5 \cdot 15

Apply exponent rule:

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

n

is even

(- 10)^{2} = 1 0^{2}

= 1 0^{2} + 4 \cdot 5 \cdot 15

Multiply the numbers:

4 \cdot 5 \cdot 15 = 300

= 1 0^{2} + 300

1 0^{2} = 100

= 100 + 300

Add the numbers:

100 + 300 = 400

= 400

Factor the number:

400 = 2 0^{2}

= 2 0^{2}

Apply radical rule:

n a^{n} = a

2 0^{2} = 20

= 20

u_{1, 2} = \frac{- ( - 10 ) \pm 20}{2 ( - 5 )}

Separate the solutions

u_{1} = \frac{- ( - 10 ) + 20}{2 ( - 5 )}, u_{2} = \frac{- ( - 10 ) - 20}{2 ( - 5 )}

u = \frac{- ( - 10 ) + 20}{2 ( - 5 )} : - 3

\frac{- ( - 10 ) + 20}{2 ( - 5 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{10 + 20}{- 2 \cdot 5}

Add the numbers:

10 + 20 = 30

= \frac{30}{- 2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{30}{- 10}

Apply the fraction rule:

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

= - \frac{30}{10}

Divide the numbers:

\frac{30}{10} = 3

= - 3

u = \frac{- ( - 10 ) - 20}{2 ( - 5 )} : 1

\frac{- ( - 10 ) - 20}{2 ( - 5 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{10 - 20}{- 2 \cdot 5}

Subtract the numbers:

10 - 20 = - 10

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

Multiply the numbers:

2 \cdot 5 = 10

= \frac{- 10}{- 10}

Apply the fraction rule:

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

= \frac{10}{10}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

u = - 3, u = 1

Substitute back

u = sec (x)

sec (x) = - 3, sec (x) = 1

sec (x) = - 3, sec (x) = 1

sec (x) = - 3 : x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

sec (x) = - 3

Apply trig inverse properties

sec (x) = - 3

General solutions for

sec (x) = - 3

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

x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

sec (x) = 1 : x = 2 πn

sec (x) = 1

General solutions for

sec (x) = 1

sec (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

Combine all the solutions

x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn, x = 2 πn

Show solutions in decimal form

x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn, x = 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for -5sec^2(x)-10sec(x)=-15 ?
The general solution for -5sec^2(x)-10sec(x)=-15 is x=1.91063…+2pin,x=-1.91063…+2pin,x=2pin