# A Guide to the Completing the Square Method

calendar_todayOctober 30, 2023 ## Introduction

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))`

## Example

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`