What is the general solution for 2cos^2(w)+5cos(w)+3=0 ?

The general solution for 2cos^2(w)+5cos(w)+3=0 is w=pi+2pin

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Popular >

2cos^2(w)+5cos(w)+3=0

Solution

2 cos^{2} (w) + 5 cos (w) + 3 = 0

Solution

w = π + 2 πn

Degrees

w = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

2 cos^{2} (w) + 5 cos (w) + 3 = 0

Solve by substitution

2 cos^{2} (w) + 5 cos (w) + 3 = 0

Let:

cos (w) = u

2 u^{2} + 5 u + 3 = 0

2 u^{2} + 5 u + 3 = 0 : u = - 1, u = - \frac{3}{2}

2 u^{2} + 5 u + 3 = 0

Solve with the quadratic formula

2 u^{2} + 5 u + 3 = 0

Quadratic Equation Formula:

For

a = 2, b = 5, c = 3

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

5^{2} - 4 \cdot 2 \cdot 3 = 1

5^{2} - 4 \cdot 2 \cdot 3

Multiply the numbers:

4 \cdot 2 \cdot 3 = 24

= 5^{2} - 24

5^{2} = 25

= 25 - 24

Subtract the numbers:

25 - 24 = 1

= 1

Apply rule

1 = 1

= 1

u_{1, 2} = \frac{- 5 \pm 1}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- 5 + 1}{2 \cdot 2}, u_{2} = \frac{- 5 - 1}{2 \cdot 2}

u = \frac{- 5 + 1}{2 \cdot 2} : - 1

\frac{- 5 + 1}{2 \cdot 2}

Add/Subtract the numbers:

- 5 + 1 = - 4

= \frac{- 4}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 4}{4}

Apply the fraction rule:

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

= - \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= - 1

u = \frac{- 5 - 1}{2 \cdot 2} : - \frac{3}{2}

\frac{- 5 - 1}{2 \cdot 2}

Subtract the numbers:

- 5 - 1 = - 6

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 6}{4}

Apply the fraction rule:

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

= - \frac{6}{4}

Cancel the common factor:

2

= - \frac{3}{2}

The solutions to the quadratic equation are:

u = - 1, u = - \frac{3}{2}

Substitute back

u = cos (w)

cos (w) = - 1, cos (w) = - \frac{3}{2}

cos (w) = - 1, cos (w) = - \frac{3}{2}

cos (w) = - 1 : w = π + 2 πn

cos (w) = - 1

General solutions for

cos (w) = - 1

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}

w = π + 2 πn

w = π + 2 πn

cos (w) = - \frac{3}{2} :

No Solution

cos (w) = - \frac{3}{2}

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

w = π + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2cos^2(w)+5cos(w)+3=0 ?
The general solution for 2cos^2(w)+5cos(w)+3=0 is w=pi+2pin