Find the value of $p$ if $px^2+2x+3p = 0$ is the quadratic equation and having equal roots

$px^2+2x+3p = 0$ is the given quadratic equation and both roots are equal for this quadratic equation. Therefore, the discriminant of this quadratic equation is equal to zero.

If $ax^2+bx+c = 0$ is the quadratic equation and having equal roots, the discriminant $b^2-4ac$ is equal to zero.

$b^2 \,– 4ac = 0$

Compare the given quadratic equation with standard form quadratic equation. Therefore $a = p$, $b = 2$ and $c = 3p$ and substitute these values in the condition.

$\implies 2^2 \,–\, 4 \times p \times 3p = 0$

$\implies 4 \,–\, 12p^2 = 0$

$\implies 4 = 12p^2$

$\implies 12p^2 = 4$

$\implies p^2 = \dfrac{4}{12}$

$\require{cancel} \implies p^2 = \dfrac{\cancel{4}}{\cancel{12}}$

$\implies p^2 = \dfrac{1}{3}$

$\implies p = \pm \sqrt{\dfrac{1}{3}}$

$\therefore \,\, p = \pm \dfrac{1}{\sqrt{3}}$

The values of $p$ are $\dfrac{1}{\sqrt{3}}$ and $-\dfrac{1}{\sqrt{3}}$.