The document derives the quadratic formula step-by-step from the quadratic equation ax2 + bx + c = 0. It starts with the quadratic equation and isolates the variables, completes the square, applies properties of exponents and square roots, and finally isolates x to arrive at the quadratic formula: x = (-b ± √(b2 - 4ac))/2a. The derivation shows the mathematical steps to transform the quadratic equation into the quadratic formula.

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 ± b 2 − 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
x + x+ =0
2
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.
b c
x + x+ =0
2
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.
b b2 b 2 ( 4a ) c
x2 + x + 2 = 2 −
a 4a 4a ( 4a ) a
• Combine the fractions
b b2 b 2 − ( 4a ) c
x2 + x + 2 =
a 4a 4a 2
© D. T. Simmons, 2009 6

7. Deriving the Quadratic Formula
• Rewrite in exponential form.
b b2 b 2 − ( 4a ) c
x2 + x + 2 =
a 4a 4a 2
b 2 − ( 4a ) c
2
 b 
 x + 2a ÷ =
  4a 2
© D. T. Simmons, 2009 7

8. Deriving the Quadratic Formula
• Apply the square root property
b 2 − ( 4a ) c
2
 b 
 x + 2a ÷ = ± 4a 2
 
b ± b 2 − ( 4a ) c
x+ =
2a 2a
© D. T. Simmons, 2009 8

9. Deriving the Quadratic Formula
• Isolate x.
b ± b − ( 4a ) c
2
b b
x+ − =− +
2a 2a 2a 2a
−b ± b 2 − ( 4a ) c
x=
2a
© D. T. Simmons, 2009 9

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