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

Given:  ax² + bx + c = 0
Prove:  x = (-b ±
Ö(b² - 4ac)) / 2a

 Step Reason ax² + bx + c = 0 1. Given x² + (b/a)x + (c/a) = 0 2. Divide by a Set d = b/(2a) 3. First step in completing the square (x + d)² = (x + b/(2a))² 4. Substitute step 3 (x + b/(2a))² = x² + (b/a)x + b²/(4a²) 5. FOIL x² + (b/a)x + b²/(4a²) = (x + b/(2a))² 6. Reverse order from step 5 x² + (b/a)x  = (x + b/(2a))² - b²/(4a²) 7. Subtract b²/(4a²) (x + b/(2a))² - b²/(4a²) + (c/a) = 0 8. Substitute step 7 into step 2. (x + b/(2a))² + (c/a) = b²/(4a²) 9. Add b²/(4a²) (x + b/(2a))² = b²/(4a²) – (c/a) 10. Subtract (c/a) = b²/(4a²) – (4ac)/(4a²) 11. Multiply (c/a) by (4a)/(4a) to get a common denominator = (b² – 4ac)/(4a²) 12. Combine numerators (x + b/(2a))  = ±Ö((b² – 4ac)/(4a²)) 13. Take square root = ±Ö(b² – 4ac)/ Ö(4a²) 14. Ö(a/b) = Öa / Öb = ±Ö(b² – 4ac)/ (2a) 15. Ö(4a²) = 2a x = -(b/(2a)) ± Ö(b² – 4ac)/ (2a) 16. Subtract b/(2a) = (-b ± Ö(b² - 4ac)) / 2a 17. Combine numerators Q.E.D.