In the following, determine whether the given quadratic equations have real roots and if so, find the roots:

$3a^2x^2+8abx+4b^2=0, a≠0$

Given:

Given quadratic equation is $3a^2x^2+8abx+4b^2=0, a≠0$.

To do:

We have to determine whether the given quadratic equation has real roots.

Solution:

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

$a=3a^2, b=8ab$ and $c=4b^2$.

The discriminant of the standard form of the quadratic equation $ax^2+bx+c=0$ is

$D=b^2-4ac$.

Therefore,

$D=(8ab)^2-4(3a^2)(4b^2)=64a^2b^2-48a^2b^2=16a^2b^2=(4ab)^2$.

As $D>0$, the given quadratic equation has real roots and the roots are

$x=\frac{-b\pm \sqrt{D}}{2a}$

$x=\frac{-8ab\pm \sqrt{(4ab)^2}}{2(3a^2)}$ 

$x=\frac{-8ab\pm 4ab}{6a^2}$ 

$x=\frac{2(-4ab\pm 2ab)}{2(3a^2)}$ 

$x=\frac{-4ab+2ab}{3a^2}$ or $x=\frac{-4ab-2ab}{3a^2}$

$x=\frac{-2ab}{3a^2}$ or $x=\frac{-6ab}{3a^2}$

$x=\frac{-2b}{3a}$ or $x=\frac{-2b}{a}$

$x=-\frac{2b}{3a}$ or $x=-\frac{2b}{a}$

The roots are $-\frac{2b}{3a}$ and $-\frac{2b}{a}$. 

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Updated on: 10-Oct-2022

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