EquationCraft

Completing the square calculator

Solve any quadratic by completing the square, with every step and the (b/2)² move spelled out. Then see why it works with tiles.

Use ^ for powers, sqrt() for roots, |x| for absolute value, and separate a system with ;.

How to complete the square

  1. Make a = 1. Divide every term by aa. 2x2+8x−10=0⇒x2+4x−5=02x^2+8x-10=0 \Rightarrow x^2+4x-5=0
  2. Move the constant. x2+4x=5x^2+4x=5
  3. Halve b and square it. (42)2=4\left(\tfrac{4}{2}\right)^2 = 4. Add it to both sides: x2+4x+4=9x^2+4x+4=9
  4. Write the perfect square. (x+2)2=9(x+2)^2 = 9
  5. Square-root both sides, with ±. x+2=±3x+2=\pm 3
  6. Solve. x=1x = 1 or x=−5x = -5

The same steps give vertex form: x2+4x−5=(x+2)2−9x^2+4x-5 = (x+2)^2-9, so the vertex is (−2,−9)(-2,-9).

See why it’s called “completing the square”

6
x²

x2+6xx^2 + 6x

Start with x² + 6x: one big x-by-x square and 6 thin x-by-1 strips.

Frequently asked questions

What does completing the square mean?
Rewriting x² + bx as a perfect square minus a constant: x² + bx = (x + b/2)² − (b/2)². Geometrically, you add the small missing corner square that turns an L-shape into a full square.
How do you complete the square when a is not 1?
Divide every term by a first (or factor a out of the x² and x terms), complete the square on what’s inside, then continue as usual.
Why add (b/2)² to both sides?
Because (x + b/2)² expands to x² + bx + (b/2)². Adding (b/2)² makes the left side a perfect square; adding it to the right keeps the equation balanced.
When should I complete the square instead of using the formula?
When you need vertex form (to graph a parabola or find its maximum or minimum), when a = 1 and b is even, or when you’re asked to derive the quadratic formula.