Question

For two real values of $n$, the equation $9x^2+nx+36=0$ has exactly one solution in $x$. What is the positive value of $n$?

Answer 1

36

Explanation:

If the quadratic expression on the LHS has exactly one root in $x$, then it must be a perfect square. Dividing 9 from both sides, we have $x^2+\frac{n}{9}x+4=0$. In order for the LHS to be a perfect square, it must factor to either $(x+2)^2=x^2+4x+4$ or $(x-2)^2=x^2-4x+4$ (since the leading coefficient and the constant term are already defined). Only the first case gives a positive value of $n$, which is $n=4\cdot9=\boxed{36}$.

Answer 2

Completing the square, you get n = 24.