Quadratic Formula: Proof

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

Proof

Formula

For a quadratic equation
ax2 + bx + c = 0
(a ≠ 0),

x = [-b ± √b2 - 4ac] / 2a.

This is the quadratic formula.

Let's see the proof of the 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.