Question

If the roots of the equations ax² + 2bx + c = 0 and $b{x}^2-2\sqrt{ac}x+b=0$ are simultaneously real, then prove that b² = ac.

Answer 1

The given equations are

…... (1)

…… (2)

Roots are simultaneously real

Then prove that.

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

Then,

And

Both the given equation will have real roots, if

…… (3)

…... (4)

From equations (3) and (4) we get

Hence,