Question:

If the quadratic equation $p x^{2}-2 \sqrt{5} p x+15=0$ has two equal roots then find the value of $p$.

Solution:

It is given that the quadratic equation $p x^{2}-2 \sqrt{5} p x+15=0$ has two equal roots.

$\therefore D=0$

$\Rightarrow(-2 \sqrt{5} p)^{2}-4 \times p \times 15=0$

$\Rightarrow 20 p^{2}-60 p=0$

$\Rightarrow 20 p(p-3)=0$

$\Rightarrow p=0$ or $p-3=0$

$\Rightarrow p=0$ or $p=3$

For p = 0, we get 15 = 0, which is not true.

∴ p ≠ 0

Hence, the value of p is 3.