In 3–5 sentences, explain why complex solutions come in conjugate pairs. Consider the structure of the quadratic formula in your response.

Answer 1

Complex solutions arise in conjugate pairs due to the nature of polynomials and the quadratic formula, which is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). When the discriminant \( (b^2 - 4ac) \) is negative, the square root term involves the imaginary unit \( i \), leading to two solutions: one with a positive and one with a negative imaginary part. This results in solutions of the form \( a + bi \) and \( a - bi \), thus demonstrating that complex roots must appear in conjugate pairs in order for the coefficients of the polynomial to remain real. This conjugate relationship ensures the overall polynomial maintains real values when expressed in standard form.