Where does the quadratic formula come from?

We're told that ax² + bx + c = 0 has solutions x = (−b ± √(b² − 4ac)) / 2a. Where does that formula actually come from?

A quadratic is easy to solve when it's a perfect square: if (x + 3)² = 16 then x + 3 = ±4, done. Completing the square is the trick of rewriting any quadratic into that solvable form.

The quadratic formula is what you get if you complete the square once on the general equation, with letters instead of numbers — so the messy work is done once, forever.

Start with ax² + bx + c = 0 (a ≠ 0). Divide by a: x² + (b/a)x + c/a = 0. Move the constant: x² + (b/a)x = −c/a.

Add (b/2a)² to both sides to complete the square: (x + b/2a)² = b²/4a² − c/a = (b² − 4ac)/4a².

Take square roots and solve: x + b/2a = ±√(b² − 4ac)/2a, so x = (−b ± √(b² − 4ac))/2a. The quantity b² − 4ac (the discriminant) decides everything: positive gives two real roots, zero one repeated root, negative no real roots.

Solve 2x² − 4x − 6 = 0. Here a = 2, b = −4, c = −6, so the discriminant is (−4)² − 4·2·(−6) = 16 + 48 = 64. Then x = (4 ± 8)/4, giving x = 3 or x = −1. Check: 2(9) − 12 − 6 = 0 ✓.

• Sign slips with −b: for b = −4, −b is +4, not −4. Write the substitution out fully before simplifying.
• Forgetting the formula needs the equation in the form ax² + bx + c = 0 first — move everything to one side before reading off a, b, c.
• Dividing by a when a might be zero: if a = 0 the equation is linear, and the formula doesn't apply.

• NCERT Class 10 Mathematics, Ch. 4 — Quadratic Equations
• Khan Academy — The quadratic formula