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

ax² + bx + c = 0              ......... (1)

-ax² + bx + c = 0            ........... (2)

Roots are simultaneously real

Let D₁ and D₂ be the discriminants of equation (1) and (2) respectively,

Then,

D₁ = (b)² - 4ac

= b² - 4ac

And

D₂ = (b)² - 4 x (-a) x c

= b² + 4ac

Both the given equation will have real roots, if D1 ≥ 0 and D2 ≥ 0.
Thus,

b² - 4ac ≥ 0             

b² ≥ 4ac                        ................. (3)

And,

b² + 4ac ≥ 0                ............... (4)

Now given that a, b, c are real number and ac ≠ 0 as well as from equations (3) and (4) we get

At least one of the given equation has real roots

Hence, proved