If $a, b, c$ are real numbers such that $ac≠0$, then show that at least one of the equations $ax^2+bx+c=0$ and $-ax^2+bx+c=0$ has real roots.

Given:

Given quadratic equations are $ax^2+bx+c=0$ and $-ax^2+bx+c=0$ and $a, b, c$ are real numbers such that $ac≠0$.

To do:

We have to show that at least one of the equations $ax^2+bx+c=0$ and $-ax^2+bx+c=0$ has real roots.

Solution:

Let $D_1$ be the discriminant of $ax^2+bx+c=0$ and $D_2$ be the discriminant of $-ax^2+bx+c=0$.

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

Therefore,

$D_1=(b)^2-4(a)(c)$

$D_1=b^2-4ac$

$D_2=(b)^2-4(-a)(c)$

$D_2=b^2+4ac$

$D_1+D_2=b^2-4ac+b^2+4ac$

$D_1+D_2=2b^2$

$D_1+D_2≥0$    (Since $b$ is real)

This implies, at least one of $D_1$ and $D_2$ is greater than or equal to zero.

Therefore, at least one of the equations $ax^2+bx+c=0$ and $-ax^2+bx+c=0$ has real roots.

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Updated on: 10-Oct-2022

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