What is the general solution for cos^2(x)+cos(x)+1=0 ?

The general solution for cos^2(x)+cos(x)+1=0 is No Solution for x\in\mathbb{R}

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cos^2(x)+cos(x)+1=0

Solution

cos^{2} (x) + cos (x) + 1 = 0

Solution

No Solution for x \in R

Solution steps

cos^{2} (x) + cos (x) + 1 = 0

Solve by substitution

cos^{2} (x) + cos (x) + 1 = 0

Let:

cos (x) = u

u^{2} + u + 1 = 0

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

u^{2} + u + 1 = 0

Solve with the quadratic formula

u^{2} + u + 1 = 0

Quadratic Equation Formula:

For

a = 1, b = 1, c = 1

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

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

Simplify

1^{2} - 4 \cdot 1 \cdot 1 : 3 i

1^{2} - 4 \cdot 1 \cdot 1

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 1 - 4

Subtract the numbers:

1 - 4 = - 3

= - 3

Apply radical rule:

- a = - 1 a

- 3 = - 1 3

= - 1 3

Apply imaginary number rule:

- 1 = i

= 3 i

u_{1, 2} = \frac{- 1 \pm 3 i}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 1 + 3 i}{2 \cdot 1}, u_{2} = \frac{- 1 - 3 i}{2 \cdot 1}

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

\frac{- 1 + 3 i}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 1 + 3 i}{2}

Rewrite

\frac{- 1 + 3 i}{2}

in standard complex form:

- \frac{1}{2} + \frac{3}{2} i

\frac{- 1 + 3 i}{2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- 1 + 3 i}{2} = - \frac{1}{2} + \frac{3 i}{2}

= - \frac{1}{2} + \frac{3 i}{2}

= - \frac{1}{2} + \frac{3}{2} i

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

\frac{- 1 - 3 i}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 1 - 3 i}{2}

Rewrite

\frac{- 1 - 3 i}{2}

in standard complex form:

- \frac{1}{2} - \frac{3}{2} i

\frac{- 1 - 3 i}{2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- 1 - 3 i}{2} = - \frac{1}{2} - \frac{3 i}{2}

= - \frac{1}{2} - \frac{3 i}{2}

= - \frac{1}{2} - \frac{3}{2} i

The solutions to the quadratic equation are:

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

Substitute back

u = cos (x)

cos (x) = - \frac{1}{2} + i \frac{3}{2}, cos (x) = - \frac{1}{2} - i \frac{3}{2}

cos (x) = - \frac{1}{2} + i \frac{3}{2}, cos (x) = - \frac{1}{2} - i \frac{3}{2}

cos (x) = - \frac{1}{2} + i \frac{3}{2} :

No Solution

cos (x) = - \frac{1}{2} + i \frac{3}{2}

No Solution

cos (x) = - \frac{1}{2} - i \frac{3}{2} :

No Solution

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

No Solution

Combine all the solutions

No Solution for x \in R

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos^2(x)+cos(x)+1=0 ?
The general solution for cos^2(x)+cos(x)+1=0 is No Solution for x\in\mathbb{R}