Honors Precalculus:  Deriving the Quadratic Formula by Completing the Square
Mt Lebanon HS 2004-5
David Kosbie


Given:  ax + bx + c = 0
Prove:  x = (-b
(b - 4ac)) / 2a

Step

Reason

ax + bx + c = 0

1. Given

x + (b/a)x + (c/a) = 0

2. Divide by a

Set d = b/(2a)

3. First step in completing the square

(x + d) = (x + b/(2a))

4. Substitute step 3

(x + b/(2a)) = x + (b/a)x + b/(4a)

5. FOIL

x + (b/a)x + b/(4a) = (x + b/(2a))

6. Reverse order from step 5

x + (b/a)x  = (x + b/(2a)) - b/(4a)

7. Subtract b/(4a)

(x + b/(2a)) - b/(4a) + (c/a) = 0

8. Substitute step 7 into step 2.

(x + b/(2a)) + (c/a) = b/(4a)

9. Add b/(4a)

(x + b/(2a)) = b/(4a) (c/a)

10. Subtract (c/a)

              = b/(4a) (4ac)/(4a)

11. Multiply (c/a) by (4a)/(4a) to get a common denominator

              = (b 4ac)/(4a)

12. Combine numerators

(x + b/(2a))  = ((b 4ac)/(4a))

13. Take square root

              = (b 4ac)/ (4a)

14. (a/b) = a / b

              = (b 4ac)/ (2a)

15. (4a) = 2a

x = -(b/(2a)) (b 4ac)/ (2a)

16. Subtract b/(2a)

  = (-b (b - 4ac)) / 2a

17. Combine numerators

Q.E.D.