Consider the following quadratic equation: x2 = 9. If asked to solve it, we would naturally take the square root of 9 and end up with 3 and -3. But what if simple square root methods won't do? What if the equation includes x raised to the first power and cannot be easily factored?

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:

Given equation: 4x2 + 13x + 7 = x + 6

EXAMPLE 1: Completing the square

STEP 1: Separate The Variable Terms From The Constant Term

Separate terms to simplify: 4x2 + 13x - x = 6 - 7

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

Divide by the term of x2 : x2 + 3x = -1/4

The method of completing the square works a lot easier when the coefficient of x2 equals 1. The coefficient in our case equals 4. Dividing 4 into each member results in x2 + 3x = - 1/4.

STEP 3: Complete The Square

The coefficient of x is divided by 2 and squared: (3 / 2)2 = 9/4

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.

The resulting 9/4 is added and subtracted: x2 + 3x + 9/4 - 9/4 = -1/4

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 x2 + 3x + 9/4.

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

Factoring x2 + 3x + 9/4 gives us (x + 3/2)2

Let's now remember a more general (x + a)2 = x2 + 2ax + a2 and use it in the current example. Substituting our numbers gives us:  x2 + 3x + 9/4 = x2 + 2*(3/2)*x + (3/2)2 = (x + 3/2)2.

STEP 5: Take The Square Root

Taking the square root: ((x + 3/2)2)1/2 = (2)1/2. x = 21/2 - 3/2 & x = -21/2 - 3/2

Finally, taking the square root from both sides gives us √(x + 3/2)2 = ±√2. Or simply x + 3/2 = ±√2. We conclude this by solving for x: X1= √2 - 3/2 and X2 = - √2 - 3/2.

EXAMPLE 2: Let's Solve One More

Given equation: 2x2 + x - 7 = x2 + 9x - 12

STEP 1: Separate The Variable Terms From The Constant Term

Separate terms to simplify: 2x2 - x2 + x - 9x = -12 + 7

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

The coefficient of x2 is equal to 1

Here, the coefficient of X2 already equals 1, so no further action needed.

STEP 3: Complete The Square

The coefficient of x is divided by 2 and squared: (-8 / 2)2 = 16

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.

The resulting 16 is added and subtracted: x2 - 8x + 16 - 16 = -5

We add and subtract 16 and can see that x2 - 8x + 16 gives us a complete square.

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

Factoring x2 - 8x + 16 gives us (x - 4)2

Since the constant term -8 is with the minus sign, we use this general form: (x - a)2 = x2 - 2ax + a2. Using our numbers gives us: x2 - 8x + 16 = x2 - 2*(4)*x + (4)2 = (x - 4)2.                              

STEP 5: Take The Square Root

Taking the square root: ((x - 4)2)1/2 = (11)1/2. x = 4 + 111/2 & x = 4 - 111/2

Finally, taking the square root from both sides gives us √(x - 4)2 = ±√11. Or simply x - 4 = ±√11. We conclude this by solving for x: X1 = 4 + √11 and X2 = 4 - √11

And there you have it!