# Quadratic Formula: Proof

How to prove the quadratic formula: formula and its proof.

## Proof

### Formula

For a quadratic equation

ax^{2} + bx + c = 0

(a ≠ 0),

x = [-b ± √b^{2} - 4ac] / 2a.

This is the quadratic formula.

Let's see the proof of the quadratic formula.

### Proof

Start from ax^{2} + bx + c = 0.

Move +c to the right side.

a ≠ 0

So divide both sides by a.

Use ax^{2} + bx

to make a perfect square trinomial.

x^{2} is x^{2}.

+[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

x^{2} + 2⋅x⋅[b/a] + (b/2a)^{2}

= (x + b/2a)^{2}

+(b/2a)^{2}

= +(b^{2})/(2a)^{2}

= +b^{2}/4a^{2}

Power of a Quotient

-c/a

= -[c/a]⋅[4a/4a]

= -(4ac)/4a^{2}

-(4ac)/4a^{2} + b^{2}/4a^{2}

= [b^{2} - 4ac]/4a^{2}

(x + b/2a)^{2} = [b^{2} - 4ac]/4a^{2}

Square root both sides.

Then

x + b/2a = ±√[b^{2} - 4ac]/4a^{2}.

Quadratic Equation: Square Root

±√[b^{2} - 4ac]/4a^{2}

= ±[√b^{2} - 4ac/√4a^{2}]

Divide Radicals

√4a^{2} = √2^{2}a^{2} = 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 = ±[√b^{2} - 4ac/2a]

Move +b/2a to the right side.

Then

x = -b/2a ± √b^{2} - 4ac/2a.

-b/2a ± √b^{2} - 4ac/2a

= [-b ± √b^{2} - 4ac]/2a

So x = [-b ± √b^{2} - 4ac] / 2a.

This is the proof of the quadratic formula.