Complete the Square Calculator: Steps, Vertex Form and Solving Quadratics

How to Complete the Square, Step by Step

1. If a ≠ 1, factor it out of the x-terms: 2x² − 8x = 2(x² − 4x).
2. Take half of the x-coefficient and square it: (−4 ÷ 2)² = 4.
3. Add and subtract that number inside the bracket: 2(x² − 4x + 4 − 4).
4. Rewrite the perfect square: 2[(x − 2)² − 4], then multiply out: 2(x − 2)² − 8.
5. Add back the constant.

Solve a Quadratic by Completing the Square

When a ≠ 1 you divide through first. 3x² + 12x = 7 becomes x² + 4x = 7/3, so (x + 2)² = 19/3 and x = −2 ± √57/3. If the number on the right is negative, you get complex roots: x² + 2x + 5 = 0 becomes (x + 1)² = −4, so x = −1 ± 2i.

Why the Quadratic Formula Works

Complete the square on ax² + bx + c = 0 in general and out falls the formula. Divide by a: x² + (b/a)x = −c/a. Add (b/2a)² to both sides: (x + b/2a)² = (b² − 4ac)/4a². Take roots: x = (−b ± √(b² − 4ac))/2a. This is the real reason the formula exists, and understanding it is why exam questions often ask for completing the square in the first place.

Vertex Form of a Parabola

y = a(x − h)² + k puts the vertex at (h, k). For y = x² − 4x + 3 you get (x − 2)² − 1, so the vertex is (2, −1). For y = −x² + 2x + 8 it is −(x − 1)² + 9 with the vertex (1, 9), a maximum because a is negative. The axis of symmetry is the line x = h.

Circles: From General Form to Centre and Radius

A circle written as x² + y² − 6x + 4y − 12 = 0 hides its centre. Complete the square twice, once for x and once for y: (x − 3)² − 9 + (y + 2)² − 4 = 12, so (x − 3)² + (y + 2)² = 25. The centre is (3, −2) and the radius is 5. The calculator also handles coefficients in front of x² and y², as in 2x² + 2y² + 8x − 4y − 2 = 0 with centre (−2, 1) and radius √6.

The Area Model: Seeing the Square

For x² + 6x, draw an x by x square and two thin 3 by x rectangles, one on each side. They almost form a larger square, with a 3 by 3 corner missing. That missing corner, 9, is the number you add to complete it. The Square tab draws this for any coefficient, which makes the process stick in a way that algebra alone does not. For x² − 7x you add 49/4, and for 3x² + 12x you factor 3 first and add 4 inside.

Our Take: What Is Overrated, Underrated and Plain Wrong

Overrated: learning it as a trick for solving quadratics. The quadratic formula is built from it.

Underrated: vertex form. Completing the square turns x² + 6x + 2 into (x + 3)² − 7, which shows the minimum −7 at x = −3 at a glance.

Plain wrong: forgetting to balance. Adding (b/2)² on one side without taking it back off or adding it to the other side changes the equation.

A Short History

Completing the square is old. Al-Khwarizmi’s algebra text from about 820 CE (the source of the word “algebra”) solves quadratics by exactly this geometric idea: add a small square to make a bigger one.

Practice Problems with Answers

1. Complete the square: x² + 6x + 2
   Answer

   (x + 3)² − 7
2. Solve x² + 6x − 7 = 0
   Answer

   x = 1 or −7
3. Vertex of y = x² − 4x + 3
   Answer

   (2, −1)
4. Centre and radius of x² + y² + 10x = 0
   Answer

   (−5, 0), r = 5
5. What do you add to x² − 8x?
   Answer

   16

Why Students Get Stuck on Completing the Square, and What a Good Calculator Should Do

Searching for help with completing the square usually starts late at night with one problem that will not work out. A quadratic will not factor, and the half-and-square trick feels like magic. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.

In short: students need to see why half of b is squared and how the vertex form appears, not just see a final number, and a calculator earns trust only when its steps can be checked.

After Complete the Square: What to Solve Next

The full list of calculators and the general math solver cover everything else.