First slide

Theory of equations

Question

$If the roots of the equation x^{2} - 2 c x + a b = 0 are real and unequal, the roots of the equation x^{2} - 2 (a + b) x + (a^{2} + b^{2} + 2 c^{2}) = 0 are$

Moderate

A

real and distinct
B

real and equal
C

real
D

imaginary

Solution

$Since x^{2} - 2 c x + a b = 0 has real and unequal roots, we have$
$c^{2} - a b > 0 (∵ D > 0)$
$\begin{array}{l} F o r x^{2} - 2 (a + b) x + (a^{2} + b^{2} + 2 c^{2}) = 0 \\ D i s c r i m i n a n t, D = 4 [(a + b)^{2} - (a^{2} + b^{2} + 2 c^{2})] \\ = 4 [2 a b - 2 c^{2}] < 0 \end{array}$
Hence, it has imaginary roots.

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