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General Studies Math: Help & Review8 chapters | 85 lessons

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Lesson Transcript

Instructor:
*David Karsner*

Perfect square trinomials are a special group of polynomials that can be factored into a very convenient pattern, making them very useful in solving equations.

Before we can get to defining a perfect square trinomial, we need to review some vocabulary.

Perfect squares are numbers or expressions that are the product of a number or expression multiplied to itself. 7 times 7 is 49, so 49 is a perfect square. *x* squared times *x* squared equals *x* to the fourth, so *x* to the fourth is a perfect square.

- Binomials are algrebraic expressions containing only two terms. Example:
*x*+ 3 - Trinomials are algebraic expressions that contain three terms. Example: 3
*x*2 + 5*x*- 6

**Perfect square trinomials** are algebraic expressions with three terms that are created by multiplying a binomial to itself. Example: (3*x* + 2*y*)2 = 9*x*2 + 12*xy* + 4*y*2

Recognizing when you have these perfect square trinomials will make factoring them much simpler. They are also very helpful when solving and graphing certain kinds of equations.

With perfect square trinomials, you will need to be able to move forwards and backwards. You should be able to take the binomials and find the perfect square trinomial and you should be able to take the perfect square trinomials and create the binomials from which it came. Any time you take a binomial and multiply it to itself, you end up with a perfect square trinomial. For example, take the binomial (*x* + 2) and multiply it by itself (*x* + 2).

(*x* + 2)(*x* + 2) = *x*2 + 4*x* + 4

The result is a perfect square trinomial.

To find the perfect square trinomial from the binomial, you will follow four steps:

Step One: Square the *a*

Step Two: Square the *b*

Step Three: Multiply 2 by *a* by *b*

Step Four: Add *a*2, *b*2, and 2*ab*

(*a* + *b*)2 = *a*2 + 2*ab* + *b*2

Let's add some numbers now and find the perfect square trinomial for 2*x* - 3*y*. For this:

*a* = 2*x**b* = 3*y*

Step One: Square the *a**a*2 = 4*x*2

Step Two: Square the *b**b*2 = 9*y*2

Step Three: Multiply 2 by *a* by *'b*

2(2*x*)(-3*y*) = -12*xy*

Step Four: Add *a*2, *b*2, and 2*ab*

4*x*2 - 12*xy* + 9*y*2

A trinomial is a perfect square trinomial if it can be factored into a binomial multiplied to itself. (This is the part where you are moving the other way). In a perfect square trinomial, two of your terms will be perfect squares. If you do not have two perfect square terms, then this trinomial is not a perfect square trinomial.

Now you should find the square root of both perfect square terms. multiply the two square roots together and then by two. You should get the positive or negative version of the other term. Once again, if this is not the case, you do not have a perfect square trinomial.

For example, in the trinomial *x*2 - 12*x* + 36, both *x*2 and 36 are perfect squares.

The square root of *x*2 is *x*, the square root of 36 is 6, and 2 times *x* (which is the same as 1) times 6 equals 12*x*/-12*x*, which does equal the other term.

*x*2 - 12*x* + 36 can be factored into (*x* - 6)(*x*- 6), also written as (*x* - 6)2.

Perfect square trinomials are often introduced in algebra courses in a section that would be entitled 'Special Products.' These polynomials are grouped this way because they have a unique pattern to factoring them. The difference of squares, the sum of cubes, and the difference of cubes are other polynomials that fall into the special products category. The unique pattern with perfect square trinomials is that their factors consist of the repetition of one binomial.

Perfect square trinomials are a vital component of the completing the square algorithm. Every quadratic equation can be written as *ax*2 + *bx* + *c* = 0, which is called the standard form.

In order to solve a quadratic equation, it is possible to add the same number to both sides of the equation; thus creating a perfect square trinomial on one side and a number on the other side of the equal sign. The number to be added to both sides of the equation to create a perfect square trinomial is the value of (*b* / 2*a*)2. The trinomial can then be written as the square of a binomial.

Using the square root property on both sides of the equation yields a linear on one side and a positive/negative number on the other making it much easier to solve. If you start with the standard form of a quadratic equation and complete the square on it, the result would be the **quadratic formula**.

Completing the square using perfect square trinomials is also helpful when manipulating the terms in the equation of a circle so that the center and radius of the circle can be easily read from the equation.

A **perfect square trinomial** is a special kind of polynomial consisting of three terms. The square roots of two of the terms multiplied by two will equal either the negative or positive version of the third term. They will factor into (*a* + *b*)(*a* + *b*) or (*a* - *b*)(*a* - *b*) where *a* and *b* are the square root of the perfect square terms. If the third term is negative, you will have (*a* - *b*)2, and if the third term is positive, it will be (*a* + *b*)2. Perfect square trinomials are used to solve equations, primarily quadratics by completing the square.

- A perfect square trinomial is a special polynomial consisting of three terms
- A perfect square trinomial is created by multiplying a binomial to itself
- Two of the terms in a perfect trinomial are perfect squares
- They can be used to solve quadratics by completing the square

Once you've finished, you should be able to:

- Describe what constitutes a perfect square trinomial
- Identify a perfect square trinomial
- Explain how to use perfect square trinomials to solve quadratics

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General Studies Math: Help & Review8 chapters | 85 lessons

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