Complex solutions come in conjugate pairs due to the nature of the coefficients in polynomial equations. When using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), if the discriminant \(b^2 - 4ac\) is negative, the square root of a negative number introduces an imaginary component. The two solutions that arise from the "±" in the formula are complex conjugates, meaning if one solution is of the form \(a + bi\), the other must be \(a - bi\). This symmetry ensures that complex roots maintain the property that coefficients of the polynomial remain real when expanded.
in 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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