Question

If p, q, r, s are real number and pr = 2(q + s) then for the equation x² + px + q = 0 and x² + rx + s = 0 which of the following statement is true?

Options

• Both equation have imaginary roots

• Both have equal roots

• At least one of the equations has real roots

• None of these

Answer 1

At least one of the equations has real roots

Explanation:

Assuming that both equation have imaginary roots,

p² – 4q < 0, r² – 4s < 0 ⇒ p² + r² – 4(q + s) < 0

But given 2(q + s) = pr

∴ p² + r² – 2pr < 0 ⇒ (p – r)² < 0

Which is not true, hence our assumption is wrong, therefore at least one of the equations has real roots.