The document derives the quadratic formula step-by-step: [1] It begins with the standard quadratic equation ax2 + bx + c = 0 and transforms it through steps such as completing the square and applying the square root property, [2] This results in the familiar quadratic formula of x = (-b ± √(b2 - 4ac)) / 2a, which gives the solutions to any quadratic equation.

1. From Whence
The Quadratic Formula?
Don Simmons
© D. T. Simmons, 2009 1

2. Deriving the Quadratic Formula
Everybody taking algebra eventually learns the quadratic
formula. But few know where it comes from.
This presentation will help you understand how to get …
From: ax 2 bx c 0
b b2 4ac
To: x
2a
© D. T. Simmons, 2009 2

3. Deriving the Quadratic Formula
• Begin with the quadratic equation…
ax 2 bx c 0
• Set the leading coefficient to one.
ax 2 bx c 0
a
b c
x2 x 0
a a
© D. T. Simmons, 2009 3

4. Deriving the Quadratic Formula
• Isolate the variables on one side of the equation by
moving the c value.
2 b c
x x 0
a a
b c
x2 x
a a
© D. T. Simmons, 2009 4

5. Deriving the Quadratic Formula
• Complete the square.
b c
x2 x
a a
2 2
b b c b
x2 x
a 2a a 2a
© D. T. Simmons, 2009 5

6. Deriving the Quadratic Formula
• Create a common denominator.
2 b b2 b2 4a c
x x
a 4a 2 4a 2 4a a
• Combine the fractions
2 b b2 b2 4a c
x x
a 4a2 4a 2
© D. T. Simmons, 2009 6

7. Deriving the Quadratic Formula
• Rewrite in exponential form.
2 b b2 b2 4a c
x x
a 4a 2 4a 2
2
b b2 4a c
x
2a 4a 2
© D. T. Simmons, 2009 7

8. Deriving the Quadratic Formula
• Apply the square root property
2
b b2 4a c
x
2a 4a 2
b b2 4a c
x
2a 2a
© D. T. Simmons, 2009 8

9. Deriving the Quadratic Formula
• Isolate x.
b b b b2 4a c
x
2a 2a 2a 2a
b b2 4a c
x
2a
© D. T. Simmons, 2009 9

10. Deriving the Quadratic Formula
• Done!
2
b b 4a c
x
2a
© D. T. Simmons, 2009 10