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