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

Given, the roots of equation $x^2-2cx+ab=0......................\left(i\right)$

are real and unequal

From equation $\left(i\right)$ $a=1, b=-2c, c=ab$

Then, the discriminant $D>0$

$b^2-4ac>0$

${\left(-2c\right)}^2-4\times 1\times ab>0$

$4c^2-4ab>0$

$c^2>ab.................\left(ii\right)$

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

$a=1, b=-2\left(a+b\right), c=a^2+b^2+2c^2=0$

The discriminant of the equation $D=0$

$D=b^2-4ac=0$

${\left[-2\left(a+b\right)\right]}^2-4\times 1\times \left(a^2+b^2+2c^2\right)=0$

$4a^2+4b^2+8ab-4a^2-4b^2-8c^2=0$

$ab-c^2=0$

From equation $\left(ii\right)$ $c^2>ab$

Now, $ab-c^2<0$

Since, the discriminant is negative

Hence, the given equation $\left(iii\right)$ has imaginary roots