 # How to solve by completing the square?

## A complete guide to solving quadratic equations by completing the square method.

We are starting with an assumption that you have learnt how to solve quadratic equations using the factoring method. In case you want a refresher on the factoring method then check-out our free math class on solving quadratic equations by factor method. This will help you revise concepts and add more insights to what you are supposed to learn ahead in this blog!

Let’s start by understanding when do we use the square method?

Quadratic equations are actually used in everyday life from measuring the area of a room to finding speed, measuring profitability etc.

‘Completing the Square’ is a method that is used for solving quadratic equations where we get restricted or limited by the factor method.

Let’s take an example equation 2x^2 – 16x + 10 = 0.

First factor the equation 2(x^2 – 8x + 5) = 0 but after this point this equation cannot be factored further. So in order to solve it, you have to resort to completing the square.

What is meant by a complete or exact square?

An expression of the form (x + y)^2 is called a complete or exact square.

Multiplying this out we obtain

(x + y)^2 = (x + y)(x + y)

= x^2 + 2xy + y^2

Similarly, (x − y)^2

Multiplying we obtain:

(x − y)^2 = (x − y)(x − y)

= x^2− 2xy + y^2

In both the expressions x^2 + 2xy + y^2 and x^2 − 2xy + y^2 are referred to as complete squares as they can be written as a single term squared, that is (x + y)^2, or (x − y)^2.

Learn how to solve a quadratic equation using completing the square method – when the coefficient of x^2 is equal to 1?

Let’s take a quadratic expression – x^2+5x-2=0

Now solve by completing the square!

Step 1: Divide the equation by coefficient of x^2.

Hence, coefficient of x^2i.ea=1.

So, equation x^2/1=5x/1-2/1=0

x^2+5x-2=0

Step 2: Move the constant term (number term) to the right side.

x^2+5x-2+2=+2

x^2+5x=2

Step 3: For complete the square, we find

(b/2)^2=(5/2)^2=25/4

Here {b=5}

Adding (b/2)^2on both sides of the equation that we get in Step 2.

x^2+5x+25/4=2+25/4

(x+5/2)^2=(8+25)/4=33/4

Step 4: Taking the square root both side

(x+5/2)=√(33/4)

(x+5/2)=±√33/2

Step 5:Solve for x:

x+5/2=+√33/2  or x+5/2=-√33/2

x=√33/2-5/2or x=(-√33)/2-5/2

x=(√33-5)/2or x=(-(√33+5))/2

x=(5.7445-5)/2 or x=(-(5.7445+5))/2

x=0.37225 or x=-5.37225

Thus, solution set: {-5.372,0.372}

Now let’s write it in the form of a complete square example like (x + y)^2 . Learn how to solve a quadratic equation using completing the square method – when the coefficient of x^2 doesn’t equal to 1?

To solve a quadratic expression where the coefficient of x^2 is not 1, students can use the completing the square method along with following an extra step first.

Suppose we are given the challenge to solve this expression by completing the square method 3x^2 − 9x + 50.

We begin by factoring out the coefficient of x^2, in this case 3.

Although 3 is not a factor of 50; we can rewrite the expression as 3 (x^2 − 3x + 50/3)

The above step has helped us convert the expression in brackets to a quadratic equation with coefficient of x^2 equal to 1 and now we need to follow the same steps as done in the solution above.

Remember that the number in the complete square will be half the coefficient of x,

so we will use (x – 3/2)^2

Let’s balance the constant by subtracting the extra constant (3/2)^2 , and putting in the constant from the quadratic expression 50/3.

3 (x^2 − 3x + 50/3) = 3 {(x – 3^2)^2 – (3/2)^2 + 50/3}

-(3/2)^2 + 50/3

– 9/4 + 50/3

-27/12 + 200/12

173/12

Now putting in together x^2 − 3x + 50/3 = (x – 3/2)^2 + 173/12

3 (x^2 − 3x + 50/3) = 3{(x – 3^2)^2 + 173/12}

If you feel confident about completing the square method for solving quadratic equations, then take this challenge and try to solve these questions:

• X^2+10x+21=0
• X^2+8x-2=0
• 3x^2+2x+1=0
• 2x^2+8x+1=0

Don’t fear and just give it a try and remember you can always refer back to our free math tutorial on solving quadratic equations using the completing the square method. Our free online math tutors at TutorEye host weekly classes to support learning for students in areas and topics that they find the most difficult.

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