Short-Answer Questions

Solve: $x^{2} - 4 a x + 4 a^{2} - b^{2} = 0$ [CBSE 2012]

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Solution

$x^{2} - 4 a x + 4 a^{2} - b^{2} = 0 \Rightarrow x^{2} - 4 a x + (2 a + b) (2 a - b) = 0 \Rightarrow x^{2} - [(2 a + b) + (2 a - b)] x + (2 a + b) (2 a - b) = 0 \Rightarrow x^{2} - (2 a + b) x - (2 a - b) x + (2 a + b) (2 a - b) = 0$
$\Rightarrow x [x - (2 a + b)] - (2 a - b) [x - (2 a + b)] = 0 \Rightarrow [x - (2 a + b)] [x - (2 a - b)] = 0 \Rightarrow x - (2 a + b) = 0 or x - (2 a - b) = 0 \Rightarrow x = (2 a + b) or x = (2 a - b)$

Hence, (2a + b) and (2a − b) are the roots of the given equation.

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