Complex solutions come in conjugate pairs due to the properties of polynomials with real coefficients. When we apply the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), the discriminant \(b^2 - 4ac\) determines the nature of the roots. If the discriminant is negative, the square root of a negative number introduces an imaginary component, leading to complex solutions. Since complex numbers involve an imaginary unit \(i\), the presence of \(\pm\) in the formula ensures that for every complex solution \(a + bi\), there is a corresponding conjugate solution \(a - bi\). Thus, these roots appear in conjugate pairs when the polynomial has real coefficients.
In 3-5 sentences, explain why complex solutions come in conjugate pairs. Consider the structure of the quadratic formula in your response.
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