Quadratic Formula: Proof

Start from ax² + bx + c = 0.

Move +c to the right side.

a ≠ 0

So divide both sides by a.

Use ax² + bx
to make a perfect square trinomial.

x² is x².

+[b/a]x is
+2 times
x times,
(+[b/a]x)/(+2⋅x), b/2a.

Write +(b/2a)².

Write the same +(b/2a)²
on the right side.

Quadratic Equation: Completing the Square

x² + 2⋅x⋅[b/a] + (b/2a)²
= (x + b/2a)²

+(b/2a)²
= +(b²)/(2a)²
= +b²/4a²

Power of a Quotient

-c/a
= -[c/a]⋅[4a/4a]
= -(4ac)/4a²

-(4ac)/4a² + b²/4a²
= [b² - 4ac]/4a²

(x + b/2a)² = [b² - 4ac]/4a²

Square root both sides.

Then
x + b/2a = ±√[b² - 4ac]/4a².

Quadratic Equation: Square Root

±√[b² - 4ac]/4a²
= ±[√b² - 4ac/√4a²]

Divide Radicals

√4a² = √2²a² = 2a

± is in front of the right side.
So you don't have to think about the sign of 2a.

Square Root

x + b/2a = ±[√b² - 4ac/2a]

Move +b/2a to the right side.

Then
x = -b/2a ± √b² - 4ac/2a.

-b/2a ± √b² - 4ac/2a
= [-b ± √b² - 4ac]/2a

So x = [-b ± √b² - 4ac] / 2a.

This is the proof of the quadratic formula.