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 $a+b$?
I want to find the positive solution of the equation, so I will try to solve it by completing the square. To do that, I need to add and subtract the same term on both sides of the equation to create a perfect square on the left side. The term I need to add and subtract is half of the coefficient of $x$ squared, which is $7^2=49$. So I rewrite the equation as $x^2+14x+49=33+49$. The left side is now a perfect square: $(x+7)^2=82$. Taking the square root of both sides, I get $x+7=\pm\sqrt{82}$. Subtracting $7$ from both sides, I get $x=-7\pm\sqrt{82}$. The positive solution is the one where I use the plus sign: $x=-7+\sqrt{82}$. This matches the form $\sqrt{a}-b$, where $a=82$ and $b=7$. Therefore, $a+b=82+7=89$.

