Completing the square is mathematically identical to transforming y=x² into y=a(x-h)²+k; to see this we need to complete the square for quadratic equations where a can be any number. You can view many examples below:

The first step is to factorise and take the a value outside the bracket. Press the buttons below to view the algebraic steps.

The transformation vertical stretch, horizontal translation, and vertical translation has been applied to y=x² to generate the same quadratic equation as above:

The quadratic equation:

You can check you working out below

Summary

Completing the square shows that all quadratic curves can be formed by applying transformation to the simplest quadratic curve y=x².