The equation $x^2+14x=33$ has two solutions. The positive solution has the form $\sqrt{a}-b$ for positive natural numbers $a$ and $b$. What is $a+b$?

Your Answer: 89

Solution: Completing the square, we add $(14/2)^2=49$ to both sides of the equation to get $x^2+14x+49=82 \Rightarrow (x+7)^2=82$. Taking the square root of both sides, we get $x+7=\sqrt{82}$ (we take the positive square root because we want the positive solution), or $x=\sqrt{82}-7$. Thus, $a=82$ and $b=7$, so $a+b=\boxed{89}$.

NoUReverse Jul 27, 2020