Completing the Square – Examples and Practice Problems

The technique of completing the square is a factoring technique that allows us to convert a given quadratic expression or equation in the form ax2+bx+c to the form a(xh)2+k. We can use this technique to simplify the process of solving equations when we have complex quadratic equations.

In this article, we will look at a summary of the technique of completing the square. Then, we will use this technique to solve some practice problems.

ALGEBRA
Formula to complete the square white background

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Learning to complete the square with examples.

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ALGEBRA
Formula to complete the square white background

Relevant for

Learning to complete the square with examples.

See examples

Summary of how to complete the square

Completing the square consists of converting a quadratic expression given in the form $latex ax^2+bx+c$ to the following form:

$latex a(x+p)^2+q$

where p and q are constants.

We can follow the steps below to complete the square of any quadratic expression or equation:

Step 1: We divide the entire expression by a when a is different from 1. In this way, we will always obtain a quadratic term with a coefficient equal to 1:

$latex x^2+bx+c$

Step 2: We take the coefficient of x (the coefficient b) and divide it by 2:

$$\left(\frac{b}{2}\right)$$

Step 3: We take the expression from step 2 and square it:

$$\left(\frac{b}{2}\right)^2$$

Step 4: We add and subtract the expression obtained in step 3 to the expression obtained in step 1:

$$x^2+bx+\left(\frac{b}{2}\right)^2-\left(\frac{b}{2}\right)^2+c$$

Step 5: We use the identity $latex x^2+2xy+y^2=(x+y)^2$ to factor the expression:

$$\left(x+\frac{b}{2}\right)^2-\left(\frac{b}{2}\right)^2+c$$

Step 6: We multiply the expression resulting from step 5 by the number by which we divided in step 1.

How to solve quadratic equations by completing the square

We can apply the method of completing the square to solve quadratic equations easily. After completing the square and obtaining an expression in the form $latex (x-h)^2+k$, we can write it as follows:

$latex (x-h)^2=-k$

Having this equation, we can take the square root of both sides. This will allow us to isolate the variable x easily.


Completing the square – Examples with answers

The method of completing the square is applied to solve the following examples. In some examples, we will only have to complete the square and in others, we will have to solve the quadratic equations.

EXAMPLE 1

Complete the square of the expression $latex x^2+2x-5$.

Since the coefficient of the quadratic term is equal to 1, we don’t have to divide the expression by any numbers initially.

We see that the coefficient b is equal to 2. Therefore, we have:

$$\left(\frac{b}{2}\right)^2=\left(\frac{2}{2}\right)^2=1$$

Adding and subtracting this value, we have:

$$x^2+2x-5=x^2+2x+1-1-5$$

Completing the square and simplifying, we have:

$latex = (x+1)^2-1-5$

$latex = (x+1)^2-6$

EXAMPLE 2

Complete the square of the expression $latex x^2+4x+10$.

We don’t have to apply the first step, since the coefficient of the quadratic term is equal to 1.

Now, we can see that the coefficient b is equal to 4. Therefore, we have:

$$\left(\frac{b}{2}\right)^2=\left(\frac{4}{2}\right)^2$$

$$=2^2$$

When we add and subtract this expression, we have:

$$x^2+4x+10=x^2+4x+2^2-2^2+10$$

Completing the square and simplifying, we have:

$latex = (x+2)^2-4+10$

$latex = (x+2)^2+6$

EXAMPLE 3

Complete the square of the expression $latex 2x^2+6x+6$.

Here, the expression has a quadratic term with a coefficient other than 1. Therefore, we can divide the entire expression by 2 to get the following

⇒ $latex x^2+3x+3$.

Given that the coefficient b equals 3, we have:

$$\left(\frac{b}{2}\right)^2=\left(\frac{3}{2}\right)^2$$

Adding and subtracting this value, we have:

$$x^2+3x+3=x^2+3x+\left(\frac{3}{2}\right)^2-\left(\frac{3}{2}\right)^2+3$$

Completing the square and simplifying, we have:

$latex = (x+\frac{3}{2})^2-\frac{9}{4}+3$

$latex = (x+\frac{3}{2})^2+\frac{3}{4}$

Since we divided the expression by 2 initially, we multiply the result by 2:

⇒  $latex 2(x+\frac{3}{2})^2+\frac{3}{2}$

EXAMPLE 4

Solve the equation $latex x^2+4x-5=0$ using the method of completing the square.

In this equation, b is equal to 4. Therefore, we have:

$$\left(\frac{b}{2}\right)^2=\left(\frac{4}{2}\right)^2$$

$$=2^2$$

Adding and subtracting this value to the quadratic equation, we have:

$$x^2+4x-5=x^2+4x+2^2-2^2-5$$

Completing the square and simplifying, we have:

$latex = (x+2)^2-4-5$

$latex = (x+2)^2-9$

Now, we can write the equation as follows:

⇒  $latex (x+2)^2=9$

Taking the square root of both sides, we have:

⇒  $latex x+2=\sqrt{9}$

⇒  $latex x+2=3$

⇒  $latex x=1$

EXAMPLE 5

Solve the equation $latex 2x^2-8x-8=0$ using the method of completing the square.

We divide the equation by 2 to obtain an equation where the coefficient of the quadratic term is equal to 1:

$latex x^2-4x-4=0$

Now, we see that the coefficient b is equal to -4. Therefore, we have:

$$\left(\frac{b}{2}\right)^2=\left(\frac{-4}{2}\right)^2$$

$$=(-2)^2$$

Adding and subtracting that value to the equation, we have:

$$x^2-4x-4=x^2-4x+(-2)^2-(-2)^2-4$$

Completing the square and simplifying, we have:

$latex = (x-2)^2-4-4$

$latex = (x-2)^2-8$

Now, we write the equation as follows:

⇒  $latex (x-2)^2=8$

Taking the square root of both sides, we have:

⇒  $latex x-2=\sqrt{8}$

⇒  $latex x=2\pm \sqrt{8}$

EXAMPLE 6

Find the solutions of the equation $latex 2x^2+12x-14=0$ using the method of completing the square.

Dividing the equation by 2, we can make the coefficient of the quadratic term equal to 1:

⇒ $latex x^2+6x-7=0$

Now, we have that the coefficient b is equal to 6. Therefore, we have:

$$\left(\frac{b}{2}\right)^2=\left(\frac{6}{2}\right)^2$$

$$=3^2$$

If we add and subtract this value to the equation, we have:

$$x^2+6x-7=x^2+6x+3^2-3^2-7$$

Completing the square and simplifying, we have:

$latex = (x+3)^2-9-7$

$latex = (x+3)^2-16$

We can write the equation as follows:

$latex (x+3)^2=16$

Taking the square root of both sides, we have:

⇒ $latex x+3=4$

⇒ $latex x=1$

EXAMPLE 7

Solve the quadratic equation $latex 3x^2-12x-3=0$ using the method of completing the square.

We start by dividing the equation by 3 to make the coefficient of the quadratic term equal to 1:

⇒ $latex x^2-4x-1=0$

We see that the coefficient b is equal to -4. Therefore, we have:

$$\left(\frac{b}{2}\right)^2=\left(\frac{-4}{2}\right)^2$$

$$=(-2)^2$$

Adding and subtracting this value to the equation, we have:

$$x^2-4x-1=x^2-4x+(-2)^2-(-2)^2-1$$

Completing the square and simplifying, we have:

$latex = (x-2)^2-4-1$

$latex = (x-2)^2-5$

We can write the equation as follows:

$latex (x-2)^2=5$

We can solve the equation by taking the square root of both sides:

⇒ $latex (x-2)=\sqrt{5}$

⇒ $latex x=2\pm \sqrt{5}$


Completing the square – Practice problems

In the following problems, you will have to complete the square of the quadratic expressions, and you will have to solve the equations using the method of completing the square.

Find the solutions to the equation $latex x^2-4x-1=0$ using the method of completing the square.

Choose an answer






Complete the square of the expression $latex x^2-3x+1$.

Choose an answer






What is the solution to the equation $latex x^2+x-1=0$ using the method of completing the square?

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Find the solution to the equation $latex x^2-8x-3=0$ using the method of completing the square.

Choose an answer






Solve the equation $latex 3x^2-6x+1=0$ using the method of completing the square.

Choose an answer







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