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?

To solve that we first need to write it as an equation where x is the number:
85 + x^2 = (x-17)^2
85 + x^2 = x^2 - 34x + 289
Now organize the equation by gathering like terms in the same side lf the equation:
You can shift 85 to the other side by subtracting 85 from both sides of the equation:

85 - 85 + x^2 =x^2 34x +289 - 85
x^2 =x^2 + 34x +204

And shift x^2 to the other side by subtracting x^2 from both sides:
x^2 - x^2 =x^2 -x^2 + 34x +204
0 = 34x + 204

Shift 34x to the other side by subtracting 34x from sides
-34x = 34x - 34x + 204
-34x = 204

And now isolate x by dividing both sides of the equation by -34
-34/-34x = 204/-34
x = -6
Thats your awnser , the number is -6
I hope you understood my brief explanation...