MathematicsGeneralQuality 88 · Exceptional

Deriving the Quadratic Formula

@author · 2026-08-21 · v1

7 min read

Start from

ax^2+bx+c=0

. Divide by

a

(assuming

a\neq 0

x^2+\frac{b}{a}x+\frac{c}{a}=0

. Complete the square by adding

\left(\frac{b}{2a}\right)^2

to both sides, factor the left, then take the square root. The familiar

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

falls out. The expression under the root,

b^2-4ac

, is the discriminant: its sign tells you whether there are two, one, or zero real solutions.

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