### Instruction: Eighty-five more than the square of a number is the same as the square of the quantity that is $17$ less than the number. What is the number?### Response: I need to find a number $x$ that satisfies the equation $85 + x^2 = (x - 17)^2$.
To solve for $x$, I will expand the right-hand side and simplify the equation.
I get $85 + x^2 = x^2 - 34x + 289$.
Subtracting $x^2$ from both sides, I get $85 = -34x + 289$.
Subtracting $289$ from both sides, I get $-204 = -34x$.
Dividing both sides by $-34$, I get $x = 6$.
To check my answer, I will plug it back into the original equation and see if it works.
I get $85 + 6^2 = (6 - 17)^2$.
Simplifying, I get $121 = 121$, which is true.
Therefore, $x = 6$ is the correct answer.