The technique of completing the square is a factoring technique that allows us to convert a given quadratic expression or equation in the form a*x*^{2}+b*x*+c to the form a(*x*–*h*)^{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.

## Formula to complete the square

The process of completing the square is used to express a quadratic expression given as $latex ax^2+bx+c$ in the following form:

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

where *p* and *q* are constants.

The simplest case of completing the square happens when we have *a*=1, that is, the quadratic term has a coefficient equal to 1. In these cases, we have:

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

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

## Completing the square – Step by step method

We can follow the steps below to complete the square of a quadratic expression. This method applies even when the coefficient *a* is different from 1.

**Step 1:** If the coefficient *a* is different from 1, we divide the entire quadratic expression by *a* to obtain an expression where the quadratic term has a coefficient equal to 1:

$latex x^2+bx+c$

**Step 2:** We divide the coefficient of *x* (the coefficient *b*) by 2:

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

**Step 3:** We square the expression obtained in step 2:

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

**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 factor the quadratic expression by applying the algebraic identity $latex x^2+2xy+y^2=(x+y)^2$:

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

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

### Solve quadratic equations by completing the square

The method of completing the square allows us to solve quadratic equations easily. When we have a quadratic expression in the form $latex (x-h)^2+k$, we can write it as follows:

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

Here, we can take the square root of both sides and easily solve for *x*.

## Completing the square – Examples with answers

### EXAMPLE 1

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

##### Solution

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 $latex x^2+4x+10$.

##### Solution

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 $latex 2x^2+6x+6$.

##### Solution

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.

##### Solution

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.

##### Solution

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.

##### Solution

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.

##### Solution

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

## See also

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