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Question:

One root of the quadratic equation ax² + bx + c = 0 is $\frac{2}{\sqrt{7} - \sqrt{5}}$ . If a and c are rational, then which of the following is true?

Explanation:

One of the roots of the given equation = $\frac{2}{\sqrt{7} - \sqrt{5}}$ × $\frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} + \sqrt{5}}$ = √7+√5

Since, a and c are rational, c/a i.e., product of the roots will also be rational.

If x is the other root of this equation, then product of roots, c/a = (√7 + √5) × x

⇒ x = $\frac{c / a}{\sqrt{7} + \sqrt{5}}$ = $\frac{c}{2 a} (\sqrt{7} - \sqrt{5})$

Now, Sum of the roots = (√7 + √5) + $\frac{c}{2 a} (\sqrt{7} - \sqrt{5})$ = $- \frac{b}{a}$

⇒ -2b = 2a(√7 + √5) + c(√7 - √5)

Hence, option (d).

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