If a and b are distinct real numbers, show that the quadratic equation $2 (a^{2} + b^{2}) x^{2} + 2 (a + b) x + 1 = 0$ has no real roots.

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Solution

The given equation is $2 (a^{2} + b^{2}) x^{2} + 2 (a + b) x + 1 = 0$ .
$∴ D = {[2 (a + b)]}^{2} - 4 \times 2 (a^{2} + b^{2}) \times 1 = 4 (a^{2} + 2 a b + b^{2}) - 8 (a^{2} + b^{2}) = 4 a^{2} + 8 a b + 4 b^{2} - 8 a^{2} - 8 b^{2} = - 4 a^{2} + 8 a b - 4 b^{2} = - 4 (a^{2} - 2 a b + b^{2}) = - 4 {(a - b)}^{2} < 0$
Hence, the given equation has no real roots.

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