# Factoring perfect square trinomials: (a + b)^2 = a^2 + 2ab + b^2 or (a - b)^2 = a^2 - 2ab + b^2

0/1
##### Introduction
###### Lessons
1. Intro to Factoring Perfect Square Polynomials
0/5
##### Examples
###### Lessons
1. Factor the perfect squares
1. $x^2 - 12x + 36$
2. $3x^2 - 30x + 75$
3. $-50x^2 + 40xy - 8y^2$
2. Find the square of a binomial:
1. $(-x - 4y)^2$
2. $(-3x^2 + 3y^2)^2$
###### Free to Join!
StudyPug is a learning help platform covering math and science from grade 4 all the way to second year university. Our video tutorials, unlimited practice problems, and step-by-step explanations provide you or your child with all the help you need to master concepts. On top of that, it's fun - with achievements, customizable avatars, and awards to keep you motivated.
• #### Easily See Your Progress

We track the progress you've made on a topic so you know what you've done. From the course view you can easily see what topics have what and the progress you've made on them. Fill the rings to completely master that section or mouse over the icon to see more details.
• #### Make Use of Our Learning Aids

###### Practice Accuracy

See how well your practice sessions are going over time.

Stay on track with our daily recommendations.

• #### Earn Achievements as You Learn

Make the most of your time as you use StudyPug to help you achieve your goals. Earn fun little badges the more you watch, practice, and use our service.
• #### Create and Customize Your Avatar

Play with our fun little avatar builder to create and customize your own avatar on StudyPug. Choose your face, eye colour, hair colour and style, and background. Unlock more options the more you use StudyPug.
###### Topic Notes
Some polynomials have common patterns, which can be factorized faster if you can recognize them. Perfect square trinomial is one of these cases.

## What is a perfect square trinomial

Let's first remember what a trinomial is. A polynomial has several terms. A trinomial (as the prefix "tri-" suggests) is a polynomial with three terms. When we're dealing with perfect squares, it means we're dealing with squaring binomials. Continue on to learn how we go about factoring a trinomial.

## How to factor perfect square trinomials

One good way to recognize if a trinomial is perfect square is to look at its first and third term. If they are both squares, there's a good chance that you may be working with a perfect square trinomial.

Let's say we're working with the following: $x^{2}+14x+49$. Is this a perfect square trinomial? Looking at the first term, we've got $x^{2}$, which is a square. The last term is $49$. It is also a square since when you multiply $7$ by $7$, you'll get $49$. Therefore $49$ can also be written as $7^{2}$. The next step to identifying if we've got a perfect square is to see if we are able to get the middle term of $14x$ when we have $x^{2}$and $7^{2}$ to work with.

In the case of a perfect square, the middle term is the first term multiplied by the last term, and then multiplied by $2$. In other words, the perfect square trinomial formula is:

$a^{2} \pm ab + b^{2}$. We're now trying to see if we can get the middle term of $2ab$.

Since we've got our $a$ term as $x$, and our $b$ term as $7$, our $2ab$ becomes $2 \bullet 7 \bullet x$. That gives us a total of $14x$, which is the middle term in $x^{2}+14x+49$! Therefore, we can rewrite the question as $(x + 7)^{2}$through factoring perfect square trinomials. You've solved a perfect square trinomial! You're now ready to apply trinomial factoring to some practice problems.

## Example problems

Question 1:

Factor the perfect square

$x^{2} - 2x + 36$

Solution:

We know that this is a perfect square, and all we're asked is to factor it. Therefore, just take a look at the first and last term and find what they are squares of. It'll give us:

$(x - 6)^{2}$

Question 2:

Factor the perfect square

$3x^{2} - 30x +75$

Solution:

Take out the common factor $3$

$3(x^{2} - 10x + 25)$

Factor the $x^{2} - 10x + 25$ and get the final answer:

$3(x - 5)^{2}$

Question 3:

Find the square of a binomial:

$(-3x^{2} + 3y^{2})^{2}$

Solution:

You can square it and it will become what we have here:

$ax^{2} - bxy +cy^{2}$

So the first term:

Square of $-3x^{2} = 9x^{4}$

The third term:

$3y^{2} = 9y^{4}$

The middle term is the multiplication of original $1^{st}$ and $2^{nd}$ term, and then times $2$

$-3x^{2} \bullet 3y^{2} = -9x^{2}y^{2}$

Then times $2$:

$-18x^{2}y^{2}$

$(9x^{4} - 18x^{2}y^{2} + 9y^{4})$

To double check your answers, this online calculator will help you factor a polynomial expression. Use it as a reference, but make sure you learn how to properly go through the steps to answering a perfect square trinomial question.

Wasn't quite sure on the concepts covered in this chapter? Perhaps you may want to go back and review how to find common factors of polynomials or how to factor by grouping. Also read up on solving polynomials with unknown coefficients, and the intro to factoring polynomials.

Ready to move on? Up next, learn how to complete the square and change a quadratic function from standard form to vertex form.