If p, q are real and p≠q, then show that the roots of the equation $(p-q)x^2+5(p+q)x-2(p-q)=0$ are real and unequal.

Given:

Given quadratic equation is $(p-q)x^2+5(p+q)x-2(p-q)=0$.

p, q are real and p≠q.

To do:

We have to show that the roots of the given quadratic equation are real and unequal.

Solution:

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

$a=(p-q), b=5(p+q)$ and $c=-2(p-q)$.

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

$D=[5(p+q)]^2-4(p-q)[-2(p-q)]$

$D=25(p+q)^2+8(p-q)^2$

$D>0$   (A positive number multiplied by a square is positive and p≠q)

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

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Updated on: 10-Oct-2022

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