Question

If the roots of the equation $x^2+2cx+ab=0$ are real unequal, prove that the equation $x^2-2(a+b)x+a^2+b^2+2c^2=0$ has no real roots.

Answer 1

The two equations are
$x^2+2cx+ab=0....\left(i\right)$ and

$x^2-2(a+b)x+a^2+b^2+2c^2=\mathrm{0......}\left(ii\right)$

Let $D_1$ and $D_2$ be the determinants of equations (i) and (ii). Then

$D_1=b^2-4ac={\left(2c\right)}^2-4\times 1\times ab=4(c^2-ab)$

$D_2=b^2-4ac={(-2(a+b\left)\right)}^2-4\times 1\times a^2+b^2+2c^2=4(c^2-ab)$

$\Rightarrow D_2=4{(a+b)}^2-4(a^2+b^2+2c^2)\Rightarrow D_2=4(2ab-2c^2)=-8(c^2-ab)$

Since the roots of equation (i) are real and unequal. Therefore,
$D_1>0$

$\Rightarrow 4(c^2-ab)=0\Rightarrow c^2-ab>0\Rightarrow -8(c^2-ab)<0$

$D_2<0$

$\therefore$ Roots of equations (ii) are not real