Where did the quadratic formula come from?

Answer 1

The quadratic formula originated from the concept of completing the square. let's take ax2 + bx + c = 0.

To complete the square, solve for x.

Subtract c.

ax2 + bx = -c.

Then divide by a [notice- if there is no a value, then a=1].

x2 + bx/a = -c/a.

Add (b/2a)2 to both sides.

x2 + bx/a + b2/4a2 = -c/a + b2/4a2

Factor/Reformat.

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

(x + b/2a)2 = [(b2-4ac) / 2a]2

Square-root both sides.

x + b/2a = ± √(b2-4ac) / 2a

Subtract b/2a.

x = -b/2a ± √(b2-4ac) / 2a

Combine terms.

x = [-b ± √(b2-4ac)] / 2a