Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 2 cos² x + 3 cos x + 1 = 0

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Recognize that the equation is quadratic in form with respect to \( \cos x \). Let \( u = \cos x \), so the equation becomes \( 2u^{2} + 3u + 1 = 0 \).

Solve the quadratic equation \( 2u^{2} + 3u + 1 = 0 \) using the quadratic formula: \( u = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \), where \( a=2 \), \( b=3 \), and \( c=1 \).

Calculate the discriminant \( \Delta = b^{2} - 4ac = 3^{2} - 4 \times 2 \times 1 \) and then find the two possible values for \( u = \cos x \).

For each value of \( \cos x \) found, determine the corresponding values of \( x \) in the interval \( [0, 2\pi) \) by using the inverse cosine function: \( x = \arccos(u) \) and considering the symmetry of cosine in the unit circle.

Write down all solutions \( x \) in \( [0, 2\pi) \) that satisfy the original equation, ensuring to check if any values of \( u \) are outside the valid range for cosine (i.e., \( -1 \leq u \leq 1 \)) and discard those if necessary.

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After determining the values of the trigonometric function, you find all angles x within the specified interval [0, 2π) that satisfy these values. This requires understanding the unit circle and the periodic nature of trigonometric functions to identify all valid solutions.

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