Question

The roots $\alpha$ and $\beta$ of the quadratic equation $px^2 + qx + r = 0$ are real and opposite signs. The roots of $\alpha(x - \beta^2)+ \beta (x - \alpha)^2 = 0$ are

• A. positive
• B. negative
• C. real and of opposite signs
• D. Imaginary

Answer 1

Answer: (C)

real and of opposite signs

Answer 2

$\alpha + \beta = - \frac{q}{p} ; \alpha \beta = \frac{r}{p}$ .Clearly $\alpha \beta < 0.$ $\alpha (x - \beta )^2 + \beta (x -\alpha)^2 = 0$ $\Rightarrow \, (\alpha + \beta)x^2 - 4 \, \alpha \beta x + \alpha \beta (\alpha + \beta) = 0$ Product of its roots = $\alpha \beta < 0$ $\therefore$ roots are real and of opposite signs.