Determine the nature of the roots of the following quadratic equations:
$(x-2a)(x-2b)=4ab$

Given:

Given quadratic equation is $(x-2a)(x-2b)=4ab$.

To do:

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

Solution:

$(x-2a)(x-2b)=4ab$

$x^2-2ax-2bx-2a(-2b)=4ab$

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

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

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

$a=1, b=-2(a+b)$ and $c=0$.

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

$D=[-2(a+b)]^2-4(1)(0)$

$D=4(a+b)^2-0$

$D=[2(a+b)]^2>0$    (Square of a number is positive)

Therefore, the roots of the given quadratic equation are real and distinct.

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Updated on: 10-Oct-2022

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