If the roots of the quadratic equation ax² + bx + c = 0 are negative to each other, then

Let the roots of the quadratic equation ax squared + bx + c = 0 be alpha and negative alpha, since they are negative to each other. The sum of the roots is alpha minus alpha, which equals 0, and by Vieta's formulas this sum equals negative b divided by a, meaning b must equal 0. The product of the roots is alpha times negative alpha, which equals negative alpha squared, and by Vieta's formulas this equals c divided by a, meaning c equals negative a times alpha squared. For the equation to genuinely have two roots that are negative to each other, alpha cannot be 0, which ensures the product is not 0 and therefore c cannot be 0. Thus, b equals 0 and c does not equal 0.

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