Question

If the roots of the equation $x^2-2cx+ab=0$ are real and unequal, then prove that the roots of $x^2-2(a+b)x+a^2+b^2+2c^2=0$ will be imaginary.

Answer 1

If the roots of $x^2-2cx+ab=0$ are real and unequal

then discriminant $D>0$

$\Rightarrow {\left(-2c\right)}^2-4ab>0$

$\Rightarrow 4c^2-4ab>0$

$\Rightarrow c^2>ab$

now in quadratic equation

$x^2-2\left(a+b\right)x+a^2+b^2+2c^2=0$

discriminant $D={\left\{-2\left(a+b\right)\right\}}^2-4\left(a^2+b^2+2c^2\right)$

$=4{\left(a+b\right)}^2-4\left(a^2+b^2+2c^2\right)$

$=4\left(2ab-2c^2\right)$

$=8\left(ab-c^2\right)$ $<$ $0$

Since discriminant is negative

$\therefore$ The roots of the given equation will be imaginary=0

x2−2(a+$