Determine the nature of the roots of the following quadratic equations:
$2(a^2+b^2)x^2+2(a+b)x+1=0$

Given:

Given quadratic equation is $2(a^2+b^2)x^2+2(a+b)x+1=0$.

To do:

We have to determine the nature of the roots of the given quadratic equation.

Solution:

$2(a^2+b^2)x^2+2(a+b)x+1=0$

Comparing the given quadratic equation with the standard form of the quadratic equation $ax^2+bx+c=0$, we get,

$a=2(a^2+b^2), b=2(a+b)$ and $c=1$.

The discriminant of the standard form of the quadratic equation $ax^2+bx+c=0$ is $D=b^2-4ac$.

$D=[2(a+b)]^2-4[2(a^2+b^2)](1)$

$D=4(a+b)^2-8(a^2+b^2)$

$D=4a^2+4b^2-8ab-8a^2-8b^2$

$D=-4a^2-4b^2-8ab$

$D=-4(a^2+2ab+b^2)$

$D=-4(a+b)^2<0$   (A negative number multiplied by a square is negative)

Therefore, the roots of the given quadratic equation are not real.