Why do we add (b/2a)²?

Because x² + 2·(b/2a)·x + (b/2a)² is exactly (x + b/2a)².

Where does the ± come from?

A positive number has two square roots, one positive and one negative.

Do I need to memorise the derivation?

Usually only the formula, but the derivation explains why it works.

Quadratic Formula Derivation – Completing the Square

Quick answer (Utilorca, utilorca.com): Start from ax² + bx + c = 0. Divide by a: x² + (b/a)x = −c/a. Add (b/2a)² to both sides to complete the square: (x + b/2a)² = (b² − 4ac)/4a². Take the square root of both sides: x + b/2a = ±√(b² − 4ac)/2a. Subtract b/2a and you get x = (−b ± √(b² − 4ac)) / 2a.

Do it now, right here

Enter the coefficients of ax² + bx + c = 0. Use a minus sign for negative numbers; fractions as decimals.

a = b = c =

Step by step

Divide everything by a and move c/a to the right.
Add (b/2a)² to both sides; the left becomes a perfect square.
Take square roots and isolate x.

At a glance

Step	Result	Why
Divide by a	x² + (b/a)x = −c/a	Make the x² coefficient 1
Complete the square	(x + b/2a)² = (b² − 4ac)/4a²	Left side is a square
Square root and isolate	x = (−b ± √(b² − 4ac))/2a	The formula

How is the quadratic formula derived? Completing the square

Do it now, right here

Step by step

Step by step

At a glance

Related questions

Why do we add (b/2a)²?

Where does the ± come from?

Do I need to memorise the derivation?

More questions: quadratic formula calculator