what are the solutions to the equation x^2-8x=24 ?

The solutions to x² - 8x  =  24:

First, get all terms to one side by subtracting 24 from both sides:

                         x² - 8x - 24  =  0

Now, try factoring this  --  but it doesn't factor nicely  -- so you can graph this, use the quadratic formula or complete the square -- I'm choosing to complete the square.

Back to moving the number back to the right side:

                        x² - 8x  =  24

Since the coefficient of the x-squared term is 1, work with the coefficient of the x-term --  -8

Completing the square is a three step process:

     --  first, divide the coefficient of the x-term by 2:  -8 / 2  =  -4

     --  second, square this number:  (-4)²  =  16

     --  third, add this number to both sides of the problem:

                  x² - 8x + 16  =  24 + 16

Factor the left side, simplify the right side:

                  (x - 4)(x - 4)  =  40

                          (x - 4)²  =  40

Take the square root of both sides:

          either  x - 4  =  sqrt(40)     or  x - 4 = - sqrt(40)

so      either  x  =  4 + sqrt(40)     or  x  =  4 - sqrt(40)

solve for x:
x^2-8 x=24

Add 16 to both sides:
x^2-8 x+16=40

Write the left hand side as a square:
(x-4)^2=40

Take the square root of both sides:
x-4=2 sqrt(10) or x-4=-2 sqrt(10)

Add 4 to both sides:
x=4+2 sqrt(10) or x-4=-2 sqrt(10)

Add 4 to both sides:
Answer: |  x=4+2 sqrt(10)       or       x=4-2 sqrt(10)