What is the general solution for 6sec^2(x)-6sec(x)=12 ?

The general solution for 6sec^2(x)-6sec(x)=12 is x= pi/3+2pin,x=(5pi)/3+2pin,x=pi+2pin

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6sec^2(x)-6sec(x)=12

Solution

6 sec^{2} (x) - 6 sec (x) = 12

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

6 sec^{2} (x) - 6 sec (x) = 12

Solve by substitution

6 sec^{2} (x) - 6 sec (x) = 12

Let:

sec (x) = u

6 u^{2} - 6 u = 12

6 u^{2} - 6 u = 12 : u = 2, u = - 1

6 u^{2} - 6 u = 12

Move

12

to the left side

6 u^{2} - 6 u = 12

Subtract

12

from both sides

6 u^{2} - 6 u - 12 = 12 - 12

Simplify

6 u^{2} - 6 u - 12 = 0

6 u^{2} - 6 u - 12 = 0

Solve with the quadratic formula

6 u^{2} - 6 u - 12 = 0

Quadratic Equation Formula:

For

a = 6, b = - 6, c = - 12

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 6 ( - 12 )}{2 \cdot 6}

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 6 ( - 12 )}{2 \cdot 6}

(- 6)^{2} - 4 \cdot 6 (- 12) = 18

(- 6)^{2} - 4 \cdot 6 (- 12)

Apply rule

- (- a) = a

= (- 6)^{2} + 4 \cdot 6 \cdot 12

Apply exponent rule:

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

n

is even

(- 6)^{2} = 6^{2}

= 6^{2} + 4 \cdot 6 \cdot 12

Multiply the numbers:

4 \cdot 6 \cdot 12 = 288

= 6^{2} + 288

6^{2} = 36

= 36 + 288

Add the numbers:

36 + 288 = 324

= 324

Factor the number:

324 = 1 8^{2}

= 1 8^{2}

Apply radical rule:

1 8^{2} = 18

= 18

u_{1, 2} = \frac{- ( - 6 ) \pm 18}{2 \cdot 6}

Separate the solutions

u_{1} = \frac{- ( - 6 ) + 18}{2 \cdot 6}, u_{2} = \frac{- ( - 6 ) - 18}{2 \cdot 6}

u = \frac{- ( - 6 ) + 18}{2 \cdot 6} : 2

\frac{- ( - 6 ) + 18}{2 \cdot 6}

Apply rule

- (- a) = a

= \frac{6 + 18}{2 \cdot 6}

Add the numbers:

6 + 18 = 24

= \frac{24}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{24}{12}

Divide the numbers:

\frac{24}{12} = 2

= 2

u = \frac{- ( - 6 ) - 18}{2 \cdot 6} : - 1

\frac{- ( - 6 ) - 18}{2 \cdot 6}

Apply rule

- (- a) = a

= \frac{6 - 18}{2 \cdot 6}

Subtract the numbers:

6 - 18 = - 12

= \frac{- 12}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 12}{12}

Apply the fraction rule:

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

= - \frac{12}{12}

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

Popular Examples

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sin (a) = \frac{2}{2}

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

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

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

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

2sin^2(2x+pi)-sin(2x+pi)-1=0

2 sin^{2} (2 x + π) - sin (2 x + π) - 1 = 0

cos(x)=9

cos (x) = 9

Frequently Asked Questions (FAQ)

What is the general solution for 6sec^2(x)-6sec(x)=12 ?
The general solution for 6sec^2(x)-6sec(x)=12 is x= pi/3+2pin,x=(5pi)/3+2pin,x=pi+2pin