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If the roots α and β (α + β ≠ 0) of the quadratic equation ax^2 + bx + c = 0 are real and of opposite sign. Then show that roots of the equation α(x - β)^2 + β(x - α)^2 = 0 are also real and of opposite sign.

mycroft holmes
272 Points
7 years ago
Product of roots is $\frac{\alpha \beta^2 + \alpha^2 \beta}{(\alpha + \beta)}=\alpha \beta<0$

This immediately means that both roots are real and of opposite signs

(if they were complex, then they are conjugates and their product is always positive)