What is the general solution for 2cos^2(x)+7cos(x)-4=0 ?

The general solution for 2cos^2(x)+7cos(x)-4=0 is x= pi/3+2pin,x=(5pi)/3+2pin

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2cos^2(x)+7cos(x)-4=0

Solution

2 cos^{2} (x) + 7 cos (x) - 4 = 0

Solution

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

Degrees

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

Solution steps

2 cos^{2} (x) + 7 cos (x) - 4 = 0

Solve by substitution

2 cos^{2} (x) + 7 cos (x) - 4 = 0

Let:

cos (x) = u

2 u^{2} + 7 u - 4 = 0

2 u^{2} + 7 u - 4 = 0 : u = \frac{1}{2}, u = - 4

2 u^{2} + 7 u - 4 = 0

Solve with the quadratic formula

2 u^{2} + 7 u - 4 = 0

Quadratic Equation Formula:

For

a = 2, b = 7, c = - 4

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 2 ( - 4 )}{2 \cdot 2}

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 2 ( - 4 )}{2 \cdot 2}

7^{2} - 4 \cdot 2 (- 4) = 9

7^{2} - 4 \cdot 2 (- 4)

Apply rule

- (- a) = a

= 7^{2} + 4 \cdot 2 \cdot 4

Multiply the numbers:

4 \cdot 2 \cdot 4 = 32

= 7^{2} + 32

7^{2} = 49

= 49 + 32

Add the numbers:

49 + 32 = 81

= 81

Factor the number:

81 = 9^{2}

= 9^{2}

Apply radical rule:

9^{2} = 9

= 9

u_{1, 2} = \frac{- 7 \pm 9}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- 7 + 9}{2 \cdot 2}, u_{2} = \frac{- 7 - 9}{2 \cdot 2}

u = \frac{- 7 + 9}{2 \cdot 2} : \frac{1}{2}

\frac{- 7 + 9}{2 \cdot 2}

Add/Subtract the numbers:

- 7 + 9 = 2

= \frac{2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

u = \frac{- 7 - 9}{2 \cdot 2} : - 4

\frac{- 7 - 9}{2 \cdot 2}

Subtract the numbers:

- 7 - 9 = - 16

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 16}{4}

Apply the fraction rule:

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

= - \frac{16}{4}

Divide the numbers:

\frac{16}{4} = 4

= - 4

The solutions to the quadratic equation are:

u = \frac{1}{2}, u = - 4

Substitute back

u = cos (x)

cos (x) = \frac{1}{2}, cos (x) = - 4

cos (x) = \frac{1}{2}, cos (x) = - 4

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

cos (x) = \frac{1}{2}

General solutions for

cos (x) = \frac{1}{2}

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

cos (x) = - 4 :

No Solution

cos (x) = - 4

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

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

Popular Examples

cot(2θ)=(sqrt(3))/3

cot (2 θ) = \frac{3}{3}

4sin^2(θ/2)+8cos(θ/2)-7=0,0<= θ<= 360

4 sin^{2} (\frac{θ}{2}) + 8 cos (\frac{θ}{2}) - 7 = 0, 0 \leq θ \leq 36 0^{\circ}

-cos(θ)+1=2cos(θ)+3

- cos (θ) + 1 = 2 cos (θ) + 3

cot^2(x)+cot(x)=0

cot^{2} (x) + cot (x) = 0

3cos(2θ)+9cos(θ)-4=-2cos(θ)

3 cos (2 θ) + 9 cos (θ) - 4 = - 2 cos (θ)

Frequently Asked Questions (FAQ)

What is the general solution for 2cos^2(x)+7cos(x)-4=0 ?
The general solution for 2cos^2(x)+7cos(x)-4=0 is x= pi/3+2pin,x=(5pi)/3+2pin