# Completing the square

### From Conservapedia

**Completing the square** is a method for solving for the roots of the general quadratic equation:

*a**x*^{2} + *b**x* + *c* = 0, where

It is first taught using equations with "friendly" numbers in place of *a*, *b*, and *c* to get the student used to the process.

- What one does is add and multiply by various carefully chosen constants to create an equation of the form:

*d*^{2}*x*^{2}+ 2*d**e**x*+*e*^{2}=*f*where*d*,*e*and*f*are constants expressed in*a*,*b*, and*c*.

- This resolves to:

- (
*d**x*+*e*)^{2}=*f*(grouping) - (take square root)
- (subtract e)
- (divide by e)

- (

By then applying the process to the general equation, we can derive the quadratic formula:

*a**x*^{2}+*b**x*+*c*= 0 (given)

- 4
*a*^{2}*x*^{2}+ 4*a**b**x*+ 4*a**c*= 0 (multiply by 4a)

- 4
*a*^{2}*x*^{2}+ 4*a**b**x*= − 4*a**c*(subtract 4ac)

- 4
*a*^{2}*x*^{2}+ 4*a**b**x*+*b*^{2}= − 4*a**c*+*b*^{2}(add b^2)

- (2
*a**x*+*b*)^{2}=*b*^{2}− 4*a**c*(group each side)

- (take square root, allow for both roots)

- (subtract b)

- (divide by 2a)

We can now determine the real or imaginary roots of any quadratic equation by simply inserting *a*, *b*, and *c* into the formula.