How to Solve Quadratics by Completing the Square

Posted on July, 2024

This blog will show you how to complete the square, and how you can use completing the square to solve other problems.

Completing the square is one of the methods students can use to solve a quadratic.

It can also help you solve problems that the formula and factorisation methods cannot.

In this blog, I have used examples that do not require a calculator, but in an exam situation, you will often be expected to use a calculator to find approximate solutions or to leave your answer in surd form. Read more about dealing with exam situations here.

How to Complete the Square to Find Solutions

The general form of a quadratic is written as such:

ax2 + bx + c = 0

Completing the square relies on the fact that we can rewrite this quadratic form in the following way:

a(x+d)2 + e = 0

Simple Example

The method for getting a quadratic into this form is as follows.

We’ll consider a simple example with an ‘a’ term of 1.

x2 + 6x + 8 = 0

Step 1:

First, you need to deal with this part of the quadratic:

x2 + 6x

Start by halving the ‘b’ term and forming the following expression:

(x + 3)2

Step 2:

You’ll notice that if you expand the expression, you’ll get:

x2 + 6x + 9

The extra ‘+ 9’ is something that we need to eliminate, so we have to take it away.

(x + 3)2 - 9

We can then rewrite the quadratic as a whole as:

(x + 3)2 - 9 + 8 = 0

(x + 3)2 - 1 = 0

Step 3:

Then we need to make our square the subject of the equation by taking ‘-1’ to the other side:

(x + 3)2 = 1

Step 4:

Remove the square by rooting both sides of the equation. Remember that when you square root, you will have a positive and negative solution:

x + 3 = 1 or x + 3 = -1

Step 5:

Finally, solve for x:

x = 1 - 3 = -2

x = -1 - 3 = -4

Thus, the solutions are:

x = -2, -4

Solving a Quadratic Equation by Completing the Square

We’ll consider a simple example with an ‘a’ term greater than 1.

2x2 + 8x + 6 = 0

Step 1:

First, deal with the quadratic terms:

2x2 + 8x

Factor out the 2:

2(x2 + 4x)

Now, halve the ‘b’ term (which is 4) and form the following expression:

2(x + 2)2

Step 2:

If you were to expand the expression (x + 2)2:

(x + 2)2 = x2 + 4x + 4

Notice that we introduced an extra ‘+4’ inside the parentheses. To balance this, we subtract it inside the parentheses and multiply by 2 (the factored-out term):

2(x2 + 4x + 4 - 4) = 2((x + 2)2 - 4) = 2(x + 2)2 - 8

Rewrite the entire quadratic:

2(x + 2)2 - 8 + 6 = 0

Simplify:

2(x + 2)2 - 2 = 0

Step 3:

Make the square term the subject of the equation by isolating it:

2(x + 2)2 = 2

Divide both sides by 2:

(x + 2)2 = 1

Step 4:

Remove the square by taking the square root of both sides of the equation. Remember that when you square root, you will have a positive and negative solution:

x + 2 = 1 or x + 2 = -1

Step 5:

Solve for x:

For x + 2 = 1:

x = 1 - 2 = -1

For x + 2 = -1:

x = -1 - 2 = -3

Thus, the solutions are:

x = -1, -3

Applying Completing the Square to Solve Other Problems

Now that we’ve seen how to apply to complete the square to solve quadratic equations, let’s explore how this method can be extended to solve other types of problems, such as finding maximum or minimum values of quadratic functions and deriving useful forms in mathematics and physics.

Finding the Vertex of a Quadratic Function

One useful application of completing the square is finding the vertex of a quadratic function. The vertex is also known as the turning point. The vertex form of a quadratic function is given by:

f(x) = a(x - h)2 + k

where (h, k) is the vertex of the parabola.

To find the vertex form by completing the square:

  1. Rewrite the quadratic function in the form ax2 + bx + c.
  2. Complete the square to express it in the vertex form.

Example:

Consider the quadratic function y = x2 + 4x + 3.

    1. Rewrite the quadratic expression:
y = x2 + 4x + 3
    1. Complete the square for the x2 + 4x term:
y = (x2 + 4x + 4) - 1 y = (x + 2)2 - 1
    1. Now, the quadratic function is in vertex form:
y = (x + 2)2 - 1

From this form, we can identify that the vertex (h, k) of the parabola is (-2, -1).

Solving Real-World Problems

Completing the square is also invaluable in solving real-world problems that involve quadratic relationships, such as determining optimal values in economics, physics, or engineering.

  • Physics: Completing the square is used in deriving the kinematic equations for motion under constant acceleration.
  • Engineering: It helps optimise designs involving quadratic cost or profit functions.

Conclusion

Completing the square is a powerful algebraic technique that not only helps solve quadratic equations but also provides insight into the structure and properties of quadratic functions.

By mastering this method, students can enhance their problem-solving skills and tackle a wider range of mathematical challenges effectively.

Whether it’s finding solutions to equations or analyzing real-world scenarios, completing the square remains a fundamental tool in the mathematical toolkit.

 

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