What is the general solution for 9sec^2(x)-9sec(x)=18 ?

The general solution for 9sec^2(x)-9sec(x)=18 is x= pi/3+2pin,x=(5pi)/3+2pin,x=pi+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

9sec^2(x)-9sec(x)=18

Solution

9 sec^{2} (x) - 9 sec (x) = 18

Solution

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = π + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

9 sec^{2} (x) - 9 sec (x) = 18

Solve by substitution

9 sec^{2} (x) - 9 sec (x) = 18

Let:

sec (x) = u

9 u^{2} - 9 u = 18

9 u^{2} - 9 u = 18 : u = 2, u = - 1

9 u^{2} - 9 u = 18

Move

18

to the left side

9 u^{2} - 9 u = 18

Subtract

18

from both sides

9 u^{2} - 9 u - 18 = 18 - 18

Simplify

9 u^{2} - 9 u - 18 = 0

9 u^{2} - 9 u - 18 = 0

Solve with the quadratic formula

9 u^{2} - 9 u - 18 = 0

Quadratic Equation Formula:

For

a = 9, b = - 9, c = - 18

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 \cdot 9 ( - 18 )}{2 \cdot 9}

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 \cdot 9 ( - 18 )}{2 \cdot 9}

(- 9)^{2} - 4 \cdot 9 (- 18) = 27

(- 9)^{2} - 4 \cdot 9 (- 18)

Apply rule

- (- a) = a

= (- 9)^{2} + 4 \cdot 9 \cdot 18

Apply exponent rule:

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

n

is even

(- 9)^{2} = 9^{2}

= 9^{2} + 4 \cdot 9 \cdot 18

Multiply the numbers:

4 \cdot 9 \cdot 18 = 648

= 9^{2} + 648

9^{2} = 81

= 81 + 648

Add the numbers:

81 + 648 = 729

= 729

Factor the number:

729 = 2 7^{2}

= 2 7^{2}

Apply radical rule:

n a^{n} = a

2 7^{2} = 27

= 27

u_{1, 2} = \frac{- ( - 9 ) \pm 27}{2 \cdot 9}

Separate the solutions

u_{1} = \frac{- ( - 9 ) + 27}{2 \cdot 9}, u_{2} = \frac{- ( - 9 ) - 27}{2 \cdot 9}

u = \frac{- ( - 9 ) + 27}{2 \cdot 9} : 2

\frac{- ( - 9 ) + 27}{2 \cdot 9}

Apply rule

- (- a) = a

= \frac{9 + 27}{2 \cdot 9}

Add the numbers:

9 + 27 = 36

= \frac{36}{2 \cdot 9}

Multiply the numbers:

2 \cdot 9 = 18

= \frac{36}{18}

Divide the numbers:

\frac{36}{18} = 2

= 2

u = \frac{- ( - 9 ) - 27}{2 \cdot 9} : - 1

\frac{- ( - 9 ) - 27}{2 \cdot 9}

Apply rule

- (- a) = a

= \frac{9 - 27}{2 \cdot 9}

Subtract the numbers:

9 - 27 = - 18

= \frac{- 18}{2 \cdot 9}

Multiply the numbers:

2 \cdot 9 = 18

= \frac{- 18}{18}

Apply the fraction rule:

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

= - \frac{18}{18}

Apply rule

\frac{a}{a} = 1

= - 1

The solutions to the quadratic equation are:

u = 2, u = - 1

Substitute back

u = sec (x)

sec (x) = 2, sec (x) = - 1

sec (x) = 2, sec (x) = - 1

sec (x) = 2 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

sec (x) = 2

General solutions for

sec (x) = 2

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 = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

sec (x) = - 1 : x = π + 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 = π + 2 πn

x = π + 2 πn

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = π + 2 πn

Graph

Popular Examples

sec^2(x)-8sec(-x)-20=0

sec^{2} (x) - 8 sec (- x) - 20 = 0

3arccos(5x)=2pi

3 arccos (5 x) = 2 π

sqrt(3)cot(3θ+pi/3)+1=0

3 cot (3 θ + \frac{π}{3}) + 1 = 0

tan(θ)= 4/13

tan (θ) = \frac{4}{13}

2cos^2(x)-cos(x)-1=0,0<= x<= 2pi

2 cos^{2} (x) - cos (x) - 1 = 0, 0 \leq x \leq 2 π

Frequently Asked Questions (FAQ)

What is the general solution for 9sec^2(x)-9sec(x)=18 ?
The general solution for 9sec^2(x)-9sec(x)=18 is x= pi/3+2pin,x=(5pi)/3+2pin,x=pi+2pin