#### Question

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.

#### Solution

The given equations are

ax^{2} + bx + c = 0 ......... (1)

-ax^{2} + bx + c = 0 ........... (2)

Roots are simultaneously real

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

Then,

D_{1} = (b)^{2} - 4ac

= b^{2} - 4ac

And

D_{2} = (b)^{2} - 4 x (-a) x c

= b^{2} + 4ac

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

Thus,

b^{2} - 4ac ≥ 0

b^{2} ≥ 4ac ................. (3)

And,

b^{2} + 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