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