Question

If p, q are real and p ≠ q, then show that the roots of the equation (p − q) x² + 5(p + q) x − 2(p − q) = 0 are real and unequal.

Answer 1

The quadric equation is

Here,

As we know that

Putting the value of

$$\begin{gathered} D={\left\{5\left(p+q\right)\right\}}^2-4\left(p-q\right)\left(-2\left(p-q\right)\right) \\ =25\left(p^2+2pq+q^2\right)+8\left(p^2-2pq+q^2\right) \\ =25p^2+50pq+25q^2+8p^2-16pq+8q^2 \\ =33p^2+34pq+33q^2 \end{gathered}$$

Since, P and q are real and , therefore, the value of .

Thus, the roots of the given equation are real and unequal.

Hence, proved