Quadratic EquationHard

Question

If $ax^{2} + bx + c = 0,a \neq 0$ be an equation with integral coefficients and $D > 0$ be its discriminant. Then the equation $b^{2}x^{2} - Dx - 4ac = 0$ must have

Options

A.Two integral roots
B.Two irrational roots
C.Two rational roots
D.At least one integral root

Solution

$f(x) = b^{2}x^{2} - Dx - 4ac$

$$\begin{matrix} f(1) = b^{2} - 4ac - D = 0 \\ \text{~other root~} = \frac{c}{a}\text{~which is also rational~} \\ \therefore\ \text{~roots of~}f(x) = 0\text{~are~}1\text{~and~}\frac{c}{a} \end{matrix}$$

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