Solve each equation for exact solutions over the interval [0, 2π).
cos² x + 2 cos x + 1 = 0

Verified step by step guidance

1. Recognize that the equation is a quadratic in terms of \( \cos x \). Rewrite the equation as \( (\cos x)^2 + 2 \cos x + 1 = 0 \).
2. Notice that the quadratic expression can be factored as \( (\cos x + 1)^2 = 0 \).
3. Set the factor equal to zero: \( \cos x + 1 = 0 \), which simplifies to \( \cos x = -1 \).
4. Recall the unit circle values where \( \cos x = -1 \) within the interval \( [0, 2\pi) \).
5. Identify the exact solution(s) for \( x \) where \( \cos x = -1 \) in the given interval.

Key Concepts

Quadratic Form in Trigonometric Equations

Many trigonometric equations can be rewritten as quadratic equations by substituting a trigonometric function (e.g., cos x) with a variable. This allows the use of algebraic methods like factoring or the quadratic formula to find solutions.

Solving for Cosine Values

After rewriting the equation in quadratic form, solving for cos x involves finding the roots of the quadratic. These roots represent the cosine values for which the original equation holds true, which can then be used to find the corresponding angles.

Finding Exact Solutions on the Interval [0, 2π)

Once the cosine values are found, the exact solutions for x are determined by identifying all angles within the interval [0, 2π) whose cosine matches those values. This often involves using the unit circle and understanding cosine symmetry.