### Instruction: 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 $a+b$?### Response: 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 $b=7$.