Solve The Equation X^2-16x+15=0 For All Solutions By Completing The Square And Include At Least The First Two Middle Steps,can You Help?

We know that (x+a)² = x² + 2ax + a², so we know that a is half the coefficient of the x term.

2a = -16
a = -16/2 = -8

If we add a² to both sides of the equation, we can complete the square.

x² - 16x + 15 = 0
x² - 16x + 64 + 15 = 64    (we have added 64 so we can complete the square)
(x-8)² + 15 = 64

Solution:
(x-8)² = 64 - 15    (subtract 15 from both sides)
(x-8)² = 49    (This may be the form you prefer for the equation with the square complete.)
x - 8 = ±√49    (take the square root of both sides)
x = 8 ± 7    (add 8 to both sides)
x = {1, 15}

Check 1
We know the factors of a quadratic are (x-root1)(x-root2)
(x-1)(x-15) = x² -15x -1x +15 = x² -16x +15    (original equation. This checks both roots at once.)

Check 2
Try the roots one at a time.
(1)2 -16(1) +15 = 0    (x=1)
1 - 16 + 15 = 0    (yes)

(15)² - 16(15) + 15 = 0    (x=15)
15(15 - 16 + 1) = 0    (yes)