Consider the following quadratic equation: ** x^{2} = 9**. If asked to solve it, we would naturally take the square root of

**and end up with**

*9***and**

*3***. But what if simple square root methods won't do? What if the equation includes**

*-3***raised to the first power and cannot be easily factored?**

*x*Fortunately, there is a method for **completing the square**. As as result, a quadratic equation can be solved by taking the square root. Let's explore this step by step together.

Say we are given the following equation:

## EXAMPLE 1: Completing the square

### STEP 1: Separate The Variable Terms From The Constant Term

Let's simplify our equation. First, separate the terms that include variables from the constant terms. Next, subtract **x** from **13x** (result is **12x**) and subtract **7** from **6** (result is **-1**).

### STEP 2: Make Sure The Coefficient Of X Squared Is Equal To 1

The method of completing the square works a lot easier when the coefficient of **x ^{2}** equals

**. The coefficient in our case equals**

*1***4**. Dividing

**into each member results in**

*4***x**.

^{2}+ 3x = - 1/4### STEP 3: Complete The Square

First we need to find the constant term of our complete square. The coefficient of **x**, which equals** 3** is divided by

**2**and squared, giving us

**9/4**.

Then we add and subtract **9/4** as shown above. Doing so does not affect our equation (**9/4 - 9/4 = 0**), but gives us an expression for the complete square **x ^{2} + 3x + 9/4**.

### STEP 4: Factor The Expression X squared + 3X + 9/4

Let's now remember a more general **(x + a) ^{2} = x^{2} + 2ax + a^{2}** and use it in the current example. Substituting our numbers gives us:

**x**

^{2}+ 3x + 9/4 = x^{2}+ 2*(3/2)*x + (3/2)^{2}=**(x + 3/2)**.

^{2}### STEP 5: Take The Square Root

Finally, taking the square root from both sides gives us **√(x + 3/2) ^{2} = ±√2**. Or simply

**. We conclude this by solving for**

*x + 3/2 = ±√2***x**:

**X**and

_{1}= √2 - 3/2**X**

_{2}= - √2 - 3/2*.*## EXAMPLE 2: Let's Solve One More

### STEP 1: Separate The Variable Terms From The Constant Term

Simplify by separating the terms with variables from constant terms. Then perform subtraction and addition on both sides of the equation.

### STEP 2: Make Sure The Coefficient Of x squared Is Equal To 1

Here, the coefficient of **X ^{2}** already equals

**1**, so no further action needed.

### STEP 3: Complete The Square

As in previous example, we find the constant term of our complete square. The coefficient of **x**, which equals** -8** is divided by

**2**and squared, giving us

**16**.

We add and subtract **16** and can see that **x ^{2} - 8x + 16** gives us a complete square.

### STEP 4: Factor The Expression X squared - 8X + 16

Since the constant term **-8** is with the minus sign, we use this general form: **(x - a) ^{2} = x^{2} - 2ax + a^{2}**. Using our numbers gives us:

**x**.

^{2}- 8x + 16 = x^{2}- 2*(4)*x + (4)^{2}= (x - 4)^{2}### STEP 5: Take The Square Root

Finally, taking the square root from both sides gives us **√(x - 4) ^{2} = ±√11**. Or simply

**. We conclude this by solving for**

*x - 4 = ±√11***:**

*x***X**and

_{1}= 4 + √11**X**

_{2}= 4 - √11And there you have it!