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.