Algebra

Polynomials

Medium

What is the condition for $x^2 + (a + b)x + ab$ , with a and b non-zero, to have real roots?

none of these

a > b

ab < 0

a < b

✅ Correct Option: 1

The expression $x^2 + (a+b)x + ab$ factors as:

$(x + a)(x + b)$

The roots are $x = -a$ and $x = -b$ , which are always real for any real $a, b$ .

This can also be confirmed using the discriminant:

$D = (a+b)^2 - 4ab$

$= a^2 + 2ab + b^2 - 4ab$

$= a^2 - 2ab + b^2$

$= (a - b)^2$

Since $(a - b)^2 \geq 0$ for all real $a, b$ , the roots are always real and no extra condition is needed.

The answer is none of these.

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Solution

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BMSAT Kozhikode Past Year Questions (Topic-Wise)

What is the condition for $x^2 + (a + b)x + ab$ , with a and b non-zero, to have real roots?