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Solving Quadratic Equations by 'Completing the Square'

Updated: May 21

Simply Maths


'Completing the Square': A Step-by-Step Guide


Quadratic equations are fundamental in algebra. While there are several ways to solve them, one of the most elegant and instructive techniques is through 'completing the square'. This method transforms the quadratic equation into a perfect square trinomial, making it easier to solve.


Before diving into the steps, let's recall a few important concepts:


Quadratic Equation:

An equation of the form

Perfect Square Trinomial: A trinomial that can be written as

where d is a constant.


Our goal is to rewrite the quadratic equation in the form

where d and e are constants.


Step-by-Step Process

Let's solve the following quadratic equation:

using the method of completing the square.


1. Move the constant term to the other side:

Start with the equation. Subtract the value for the constant, in this case 5, from both sides to isolate the quadratic and linear terms:





2. Identify the coefficient of 𝑥x and find half of it:

The coefficient of x is 6. Half of 6 is 3. We will need this


3. Square the half-coefficient and add it to both sides:

Square the half-coefficient:

Add 9 to both sides of the equation:

Simplifying the right side...

4. Rewrite the left side as a perfect square trinomial:

Using our half-coefficient from earlier, the left side of the equation can be written as:

So, the equation now becomes:

5. Solve for 𝑥x by taking the square root of both sides:

Take the square root of both sides to solve for x:

This results in two possible solutions:

So, the solutions are x = −1 and x = −5.


Practice this with lots of quadratic equations (there is a worksheet for you to download below) until you are comfortable with the process - it does get easier!


 

For those interested, we can use the general form of completing the square to derive the Quadratic Formula...

This method not only provides a pathway to solving quadratic equations but also gives insight into the derivation of the quadratic formula. By mastering completing the square, you gain a deeper understanding of the algebraic structures underlying quadratic equations.


 

Completing the square
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