What is the general solution for 3sec^2(x)-3sec(x)=6 ?

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

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3sec^2(x)-3sec(x)=6

Solution

3 sec^{2} (x) - 3 sec (x) = 6

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

3 sec^{2} (x) - 3 sec (x) = 6

Solve by substitution

3 sec^{2} (x) - 3 sec (x) = 6

Let:

sec (x) = u

3 u^{2} - 3 u = 6

3 u^{2} - 3 u = 6 : u = 2, u = - 1

3 u^{2} - 3 u = 6

Move

6

to the left side

3 u^{2} - 3 u = 6

Subtract

6

from both sides

3 u^{2} - 3 u - 6 = 6 - 6

Simplify

3 u^{2} - 3 u - 6 = 0

3 u^{2} - 3 u - 6 = 0

Solve with the quadratic formula

3 u^{2} - 3 u - 6 = 0

Quadratic Equation Formula:

For

a = 3, b = - 3, c = - 6

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 3 ( - 6 )}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 3 ( - 6 )}{2 \cdot 3}

(- 3)^{2} - 4 \cdot 3 (- 6) = 9

(- 3)^{2} - 4 \cdot 3 (- 6)

Apply rule

- (- a) = a

= (- 3)^{2} + 4 \cdot 3 \cdot 6

Apply exponent rule:

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

n

is even

(- 3)^{2} = 3^{2}

= 3^{2} + 4 \cdot 3 \cdot 6

Multiply the numbers:

4 \cdot 3 \cdot 6 = 72

= 3^{2} + 72

3^{2} = 9

= 9 + 72

Add the numbers:

9 + 72 = 81

= 81

Factor the number:

81 = 9^{2}

= 9^{2}

Apply radical rule:

9^{2} = 9

= 9

u_{1, 2} = \frac{- ( - 3 ) \pm 9}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 9}{2 \cdot 3}, u_{2} = \frac{- ( - 3 ) - 9}{2 \cdot 3}

u = \frac{- ( - 3 ) + 9}{2 \cdot 3} : 2

\frac{- ( - 3 ) + 9}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{3 + 9}{2 \cdot 3}

Add the numbers:

3 + 9 = 12

= \frac{12}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= 2

u = \frac{- ( - 3 ) - 9}{2 \cdot 3} : - 1

\frac{- ( - 3 ) - 9}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{3 - 9}{2 \cdot 3}

Subtract the numbers:

3 - 9 = - 6

= \frac{- 6}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 6}{6}

Apply the fraction rule:

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

= - \frac{6}{6}

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

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Frequently Asked Questions (FAQ)

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