Question

"Completing the square involves dividing the coefficient on x by 2 and squaring the result"
but why

Answer 1

one reason we complete the square is to solve a quadratic equation that cannot be factored.

An example is as follows

solve for x

x^2 - 2x - 7 = 0
x^2 - 2x = 7
divide -2 by 2 and square the results
(x - 1)^2 - 1 = 7
we needed to subtract the 1 to make the left side equal to x^2 - 2x
(x - 1)^2 = 8
now find the square root of both sides to solve for x

Answer 2

Let's work backwards. Here's one part of our completing the square answer.
(x+a)^2

Expand this...

(x+a)(x+a)
= x^2 + ax + ax + a^2

Simplifies to:
= x^2 + 2ax + a^2

See how the coefficient (a) is double the number we had in our answer?