When does $(p+1)x^2+4px+9=0$ have a repeated root?

When does $(p+1)x^2+4px+9=0$ have a repeated root?

Question

Find the values of $p$ for which the equation $(p+1)x^2+4px+9=0$ has equal roots.

Suggestion

Find the values of $p$ for which the equation $(p+1)x^2+4px+9=0$ has equal roots.

When does the quadratic equation $ax^2 + bx + c = 0$ have equal roots?

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The red curve here is $y = (p+1)x^2+4px +9$.

What will the curve look like when the equation has equal roots?

Can you find the required values of $p$?

Solution

A quadratic equation has equal roots if and only if its discriminant is zero.

So here we require

$(4p)^2 - 4(9(p+1)) = 0$

. Dividing by

$4$

gives

$$\begin{align*} 4p^2 - 9(p+1) &=0 &\iff\quad&&4p^2 - 9p - 9 = 0 \\ &&\iff\quad&&(p - 3)(4p + 3) = 0 \\ &&\iff\quad&&p = 3 \ \text{or} \ p=-\tfrac{3}{4}. \end{align*}$$

So for these two values of $p$ (and no other values), the quadratic equation has equal roots.