Question

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

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Hint: Here we go through the properties of the quadratic equation as we know when the roots of the quadratic equation are equal then their discriminant must be equal to zero. So we equate discriminant of this equation equal to zero for finding the value of p.

We know that if in quadratic equation $a{x^2} + bx + c = 0$ when the two roots are equal then its discriminant is equal to zero I.e. ${b^2} - 4ac = 0$.
Now in the question the given quadratic equation is $p{x^2} - 2\sqrt 5 px + 15 = 0$.
By equating it with the general quadratic equation we get a=p, b$= - 2\sqrt 5 p$ and c=15.
Now we will calculate its discriminant by formula ${b^2} - 4ac = 0$.
$\Rightarrow {\left( { - 2\sqrt 5 p} \right)^2} - 4 \times p \times 15 = 0 \\ \Rightarrow 20{p^2} - 60p = 0 \\$
$\Rightarrow 20p(p - 3) = 0$
$p \ne 0$ As it makes a coefficient of ${x^2} = 0$.