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

Given:

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

To do:

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

Solution:

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

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

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

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

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

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

$D=a^2+b^2+c^2+2ab+2bc+2ca-4ab-4ac$

$D=a^2+b^2+c^2-2ab+2bc-2ca$

$D=(-a+b+c)^2$    (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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