Quadratic Formula Proof

Quadratic Formula: PROOF −𝑏± 𝑏 2 −4𝑎𝑐

If 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, Then 𝑥 = 2𝑎 ________________________________________________________________________________________ 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 Complete the square. Move the constant to the other side.

𝑎𝑥 2 + 𝑏𝑥 = −𝑐 Remove the coefficient of x2.

𝑎𝑥 2 𝑏𝑥 −𝑐 + = 𝑎 𝑎 𝑎 𝑥2 +

𝑏𝑥 𝑐 = − 𝑎 𝑎

Add new constant to each side. (1/2 the coefficient of x)2.

𝑏𝑥 𝑥 + + 𝑎

2

1 𝑏 2 𝑎

2

𝑐 = − + 𝑎

1 𝑏 2 𝑎

2

Simplify.

𝑥2 +

𝑏𝑥 𝑏2 𝑐 𝑏2 + = − + 𝑎 4𝑎2 𝑎 4𝑎2

𝑥2 +

𝑏𝑥 𝑏2 −4𝑎𝑐 + 𝑏 2 + = 𝑎 4𝑎2 4𝑎2 Perfect square.

𝑏 2𝑎

𝑥+

2

=

𝑏 2 − 4𝑎𝑐 4𝑎2

Properties of square roots. Take the square root of each side.

𝑥+

𝑏

2

2𝑎

𝑏 2 − 4𝑎𝑐

=±

4𝑎 2

Simplify.

𝑥+

𝑏 2𝑎

=

± 𝑏 2 − 4𝑎𝑐 2𝑎

Solve for x.

𝑥=

± 𝑏 2 − 4𝑎𝑐 𝑏 − 2𝑎 2𝑎

The QUADRATIC FORMLA.

𝑥=

−𝑏 ± 𝑏 2 − 4𝑎𝑐 2𝑎

Proof complete.

-Sgt Ryals, USMC