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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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Last updated date: 13th Jul 2024
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Answer
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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.

Complete step-by-step answer:
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 \\
 $
Now take 20p as common we get,
$ \Rightarrow 20p(p - 3) = 0$
  When p−3=0 then p=3 or p=0
$p \ne 0$ As it makes a coefficient of ${x^2} = 0$.
Hence, p=3 is the correct answer.

Note: Whenever we face such a question the key concept for solving the question is first point out the hint that is given in the question here in this question the hint is the roots are equal. By this hint we have to think about that case when the roots of the quadratic equation are equal. Then apply that case for finding the value of an unknown term.