First slide

Theory of equations

Question

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

Moderate

A

real and distinct
B

real and equal
C

real
D

imaginary

Solution

$c^{2} - a b > 0$
$For x^{2} - 2 (a + b) x + (a^{2} + b^{2} + 2 c^{2}) = 0$
$Discriminant, D = 4 [(a + b)^{2} - (a^{2} + b^{2} + 2 c^{2})]$
$= 4 [2 a b - 2 c^{2}] < 0$
Hence, it has imaginary roots

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