If P ( Q − r ) x 2 + Q ( r − P ) x + r ( P − Q ) = 0 has equal roots then Q 2 = (where P , Q , r ∈ R )

Question

If P ( Q − r ) x 2 + Q ( r − P ) x + r ( P − Q ) = 0 has equal roots then Q 2 ​ = (where P , Q , r ∈ R )

TestHub · Accepted Answer

The equation has equal roots, so its discriminant is zero. Also, observe that $x = 1$ is a root.

Since $x = 1$ is a root, the product of roots is $1 \times 1 = 1$ .

Thus, $\frac{r ( P - Q )}{P ( Q - r )} = 1$ .

$2 P r = PQ + r Q$

Divide by $PQR$ : $\frac{2}{Q} = \frac{1}{r} + \frac{1}{P}$ .

Mathematics - Quadratic Equation Question with Solution | TestHub