Complex solutions come in conjugate pairs due to the nature of the coefficients in polynomial equations, specifically when they are real numbers. According to the quadratic formula, the solutions of a quadratic equation \( ax^2 + bx + c = 0 \) are given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). If the discriminant \( b^2 - 4ac \) is negative, the square root term becomes imaginary, resulting in solutions of the form \( x = \frac{-b}{2a} \pm \frac{\sqrt{|b^2 - 4ac|}}{2a}i \), where the two solutions differ only by the sign in front of the imaginary part. Consequently, these solutions, which are complex numbers, are conjugates of each other, as one has a positive imaginary component and the other has a negative imaginary component.
n 3-5 sentences, explain why complex solutions come in conjugate pairs. You can use the quadratic formula to assist with your answer.
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