A Guide to the Completing the Square Method

calendar_todayOctober 30, 2023
A Guide to the Completing the Square Method


The Completing the Square Method is a widely-used mathematical technique for solving quadratic equations, as well as for graphing and optimising quadratic functions. This article aims to provide an easy-to-understand, in-depth explanation of how to use this method.

What is a Quadratic Equation?

A quadratic equation is any equation that can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable you’re solving for.

Steps for Completing the Square

Step 1: Isolate the Quadratic Term and Linear Term

Start by moving the constant term c to the other side of the equation:

ax^2 + bx = -c

Step 2: Factor Out Common Terms

If a is not 1, factor it out of the first two terms:

a(x^2 + (b/a)x) = -c

Step 3: Complete the Square

Add and subtract the same value to make it a perfect square trinomial:

a(x^2 + (b/a)x + ((b/2a)^2) - ((b/2a)^2)) = -c

This simplifies to:

a((x+(b/2a))^2 - ((b/2a)^2)) = -c

Step 4: Solve for x

Expand the equation and solve for x to find the roots of the equation:

x = -(b/2a) ± sqrt((b^2-4ac)/(4a^2))


Let’s solve the equation x^2 + 4x - 5 = 0.

  1. Isolate the quadratic and linear terms: x^2 + 4x = 5
  2. Factor out common terms: No common terms other than 1 here.
  3. Complete the Square: (x + 2)^2 - 4 = 5
  4. Solve for x: x = -2 ± sqrt(9)
    x = -2 ± 3
    x = -5, 1