Question

In 3–5 sentences, explain why complex solutions come in conjugate pairs. Consider the structure of the quadratic formula in your response.(4 points)

Answer 1

For a quadratic with real coefficients, the quadratic formula gives roots x = (-b ± √(b^2 − 4ac)) / (2a). If the discriminant b^2 − 4ac is negative, √(b^2 − 4ac) is purely imaginary, so the two roots differ only by the sign of that imaginary part. Complex conjugation commutes with addition and division, so changing the sign of the imaginary part produces the conjugate number. More generally, because coefficients are real, taking the conjugate of a root yields another root, so nonreal roots come in conjugate pairs.