Derivation of the Quadratic Formula

• The following shows how the method of Completing the Square can be used to derive the Quadratic Formula. Start with the general

quadratic equation.

Move the constant to the other side.

2 0ax bx c

Divide by a, the leading coefficient.

2ax bx c

2 b cx x

a a

Complete the square.

Add the value to both sides of the equation.

2 b cx x

a a

1

2 2

b b

a a

2 2

22 4

b b

a a

2 22

2 24 4

b b b cx x

a a a a

Factor the left side and simplify the right side.

2 22

2 24 4

b b b cx x

a a a a

2 2

2

4

2 4 4

b b c ax

a a a a

2

2 2

4

4 4

b ac

a a

2

2

4

4

b ac

a

Complete the problem using the square root method of solving a quadratic equation.

2 2

2

4

2 4

b b acx

a a

2

2

4

2 4

b b acx

a a

2 4

2 2

b b acx

a a

2 4

2 2

b b acx

a a

2 4

2 2

b b acx

a a

2 4

2

b b acx

a

Quadratic Formula

2 4

2

b b acx

a

The expression under the radical is called the Discriminant.

2 4b ac