Complex solutions appear in conjugate pairs as a consequence of the quadratic formula: x = (-b ± sqrt(b² - 4ac)) / 2a. When the discriminant (b² - 4ac) is negative, it leads to the emergence of imaginary numbers. The square root of a negative number produces both a positive and a negative imaginary component, resulting in solutions of the form a + bi and a - bi, which are complex conjugates. These conjugate pairs ensure that the coefficients of the quadratic equation remain real.
Rewrite this: Complex solutions come in conjugate pairs due to the structure of the quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / 2a. When the discriminant (b^2 - 4ac) is negative, it results in an imaginary number. Since the square root of a negative number gives both a positive and a negative imaginary component, the solutions are of the form a + bi and a - bi, which are complex conjugates. These conjugate pairs ensure that the coefficients of the quadratic equation remain real.
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