First slide

Theory of equations

Question

If a $\neq$ b, then the roots of the equation $2 (a^{2} + b^{2}) x^{2} + 2 (a + b) x + 1 = 0$ are

Moderate

A

real and distinct
B

real and equal
C

imaginary
D

None of the above

Solution

The given equation is $2 (a^{2} + b^{2}) x^{2} + 2 (a + b) x + 1 = 0$
Now,
$\begin{array}{l} D = 4 (a + b)^{2} - 8 (a^{2} + b^{2}) \\ = - 4 (a^{2} + b^{2} - 2 ab) = - 4 (a - b)^{2} < 0 (∵ a - b \neq 0) \end{array}$
Hence, the roots of the given equation are imaginary.

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