Question

If a, b, c are real numbers such that ac ≠ 0, then show that at least one of the equations ax2 + bx + c = 0 and −ax² + bx + c = 0 has real roots.

Answer 1

The given equations are

…... (1)

…… (2)

Roots are simultaneously real

Let be the discriminants of equation (1) and (2) respectively,

Then,

And

Both the given equation will have real roots, if.
Thus,
$b^2-4ac\ge 0$
$b^2\ge 4ac$ ...... (3)
And,

…... (4)

Now given that are real number and as well as from equations (3) and (4) we get

At least one of the given equation has real roots

Hence, proved