Roots of equation ax^2-2bx+c=0 are n and m, then the value of ban^2+c+bam^2+c is

D. b/c. Step 1: n is a root, so an^2-2bn+c=0, i.e. an^2+c=2bn; similarly am^2+c=2bm. Step 2: The expression is b/2bn+b/2bm=n+m/2nm. Step 3: n+m=2b/a and nm=c/a, so the value is 2b/a/2c/a=b/c. Check: a=1, b=3, c=8: x^2-6x+8=0 has roots 2 and 4, and 3/12+3/24=3/8=b/c. Trap: (B) b/a is half the sum of the roots, n+m/2; it appears if you forget to divide by nm.

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Roots of equation $a x^{2} - 2 b x + c = 0$ are n and m, then the value of $\frac{b}{a n ^{2} + c} + \frac{b}{a m ^{2} + c}$ is

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Answer: Option D — $\frac{b}{c}$

Detailed Solution

n is a root, so $a n^{2} - 2 bn + c = 0$ , i.e. $a n^{2} + c = 2 bn$ ; similarly $a m^{2} + c = 2 bm$ .

The expression is $\frac{b}{2 bn} + \frac{b}{2 bm} = \frac{n + m}{2 nm}$ .

$n + m = \frac{2 b}{a}$ and $nm = \frac{c}{a}$ , so the value is $\frac{2 b / a}{2 c / a} = \frac{b}{c}$ .

✓ Check

a = 1

b = 3

c = 8

x^{2} - 6 x + 8 = 0

has roots 2 and 4, and

\frac{3}{12} + \frac{3}{24} = \frac{3}{8} = \frac{b}{c}

⚠ Common trap

(B)

\frac{b}{a}

is half the sum of the roots,

\frac{n + m}{2}

; it appears if you forget to divide by

nm

Quick tip

$a n^{2} + c = 2 bn \Rightarrow \frac{1}{2} (\frac{1}{n} + \frac{1}{m}) = \frac{b}{c}$

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Quadratic Equations — Preparation

This question tests your understanding of Quadratic Equations, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A quadratic equation has the form ax^2 + bx + c = 0, where a is not zero. This topic covers solving quadratics by factorisation and by the quadratic formula, the nature of roots using the discriminant b^2 - 4ac, the sum (-b/a) and product (c/a) of roots, and forming an equation from its roots. In aptitude exams, a common question type gives two quadratic equations and asks you to compare their roots (whether x > y, x < y and so on). In engineering and MCA entrance exams, questions also test conditions on roots, common roots, and symmetric expressions such as the sum of squares of roots. Using the sum and product of roots is usually far faster than finding the roots themselves. Practise the quadratic equation questions below, each with a step-by-step solution.

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