Solve the quadratic function ax^2 + bx + c = 0 into the quadratic formula

What do you use to solve for x? Do you use the quadratic formula or do you complete the square? Actually, the quadratic formula is proven by completing the square with variables instead of numbers. So, just complete the square. If you already know how to solve for these functions then it stands to reason that you can for any arbitrary quadratic.

You start out with the quadratic:
ax² + bx + c = 0

You want to divide by 'a'
x² + (b/a)x + c/a = 0

Put the constant on the other side
x² + (b/a)x = −c/a

Complete the square...
x² + (b/a)x + (b/2a)² = −c/a + (b/2a)²

Factor the left side:
(x + (b/2a))² = −c/a + (b/2a)²

Simplify:
(x + (b/2a))² = (b/2a)² − c/a

(x + (b/2a))² = b²/(4a²) − c/a

(x + (b/2a))² = b²/(4a²) − (4ac)/(4a²)

(x + (b/2a))² = (b² − 4ac) / (4a²)

Square root to isolate 'x'
x + (b/2a) = ±√[ (b² − 4ac) / (4a²) ]

x + (b/2a) = ±√[ b² − 4ac ] / (2a)

x = −(b/2a) ±√[ b² − 4ac ] / (2a)

x = [ −b ±√( b² − 4ac ) ] / (2a)

And you are left with the quadratic formula

answered Mar 19, 2015 by Level 3 User (2,240 points)
reshown Mar 28, 2015 by Rod