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Solving a Quadratic Equation by Completing the Square

If a and b ar...

Question

If a and b are the odd integers, then find nature of roots of the equation
$2 a x^{2} + (2 a + b) x + b = 0$ , where a is not equal to zero.

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Solution

We have,
$\Rightarrow 2 a x^{2} = (2 a + b) x + b = 0$
$\Rightarrow$ Discriminate $= {(2 a + b)}^{2} - 8 a b$
$= 4 a^{2} + b^{2} + 4 a b - 8 a b$
$= 4 a^{2} + b^{2} - 4 a b$
$= {(2 a - b)}^{2}$
As discriminate is perfect square and $a$ and $b$ are integers, hence roots will be rational.

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