In this article, we will discuss how to find the **exact value** of a perfect-*square trinomial* or **trinomial equation**. The trick is to use the **factoring method**.

## Identifying the term with an odd number of occurrences of X

When determining the term with an even number of occurrences of X, there are *two main factors* to take into account.

The first is the term’s magnitude or size, and the second is how many times the expression has appeared in your equation.

When having a large expression such as (X + 5)2 + 10 = 35 + 25 + 15 = 55, having the *addition appear twice gives* you a term with five times the magnitude of X, so you would look for a term with an even number of occurrences of X.

When finding a perfect-square Trinomial, remember that 3 multiplied by **3 equals 6**, so in order for there to be an **addition appearing twice**, there needed to be an equation involving 6.

## Determining the difference between the term with an odd number of occurrences of X and the square root of 2

The term with an odd number of occurrences of X is called an odd term and the square root of 2 is called a positive term.

When preparing a Trinomial Expression, the first step is to determine if there are even or even terms or if the term has an odd number of occurrences of X.

If there are even terms, then the ** middle term must** be added to create a perfect-square Trinomial. The

*total amount*of

*squares required*can be found by adding up all the angles in the parentheses.

If there are even terms but one has an odd number of occurrences of X, then the middle term must be added again to create a perfect-square Trinomial with an Even Number of Angles.

This process is repeated until all angles in the expression have been accounted for.

## Calculating all terms that have an odd number of occurrences of X

The trick to finding all the *odd terms* of an expression is to add together the exponents of each term. For example, the expression 2 + 2•3 + 4 + 5•6 + 7 + 8 adds up to 12, so we have to add the exponents.

The same rule applies to Trinomials, except that we have to subtract off the top term. The rule for Trinominales is that they must have only even terms, so we must subtract off 1 from each term.

The trick to finding all the perfect-square Trinominals is to divide by 2.

## Calculating all terms that do not have an odd number of occurrences of X

This is one of the most *common mistakes made* when trying to solve Trinomial equations. The easiest way to calculate all of the non-X terms is to multiply by X itself!

This can be done easily by adding the **two sides** of the equation, but in order for this to work, there must be an even number of terms. In order for this to be true, there must be an odd number of terms because if there were an even amount of them, then 1 + 1 + 1 + 1 + **1 would equal 2**, which doesn’t exist.

So, in order for this formula to work, there must be an addition on each side! Luckily for us, computers have a way to do this for us.

## Finding the difference between each term with an odd number of occurrences of X and its corresponding squared root value

The trick to finding the square root of a Trinomial is to find the difference between each term with an even number of occurrences of X and its corresponding term with an odd number of occurrences.

The square root of an Trinomial can be found by adding up all the terms with an even number of occurrences and finding their square. The term with the higher value will be the square root, and its corresponding term will be decreased in size.

This is done by using the *2 × 2 matrix introduced earlier* in this article. In this matrix, side A represents the term with an even number of occurrences, side B represents its correspondingtermwithanoddnumberofoccurrencesthatnumberofweeks, and side C represents increased size.

When finding the difference between two values, it is important to **know whether one value** is larger or smaller than the other.

## Coming up with a final value to add to each term with an even number or occurrence to make it a perfect square trinomial

This is a *difficult math problem* to solve, so most students will not bother. However, the solution can be found in the bullet point below.

The trinomial equation can be rewritten as a special case of the perfect-square-trinomial equation. The new equation has **three terms instead** of two, but the way to add them up is still the same.

When **solving trinomial equations**, many times what students do is *take one term* of each type and add them together, and then take the other term of each type and add together. This gives them an answer to simplify the third term on the equation.

## Understanding how to simplify perfect square trinomial expressions

When a **perfect square trinomial expression** has more than one variable, it is easy to simplify by *adding additional variables*.

For example, the variable y can be replaced by the value of x in the equation. The same happens with z, the variable for t, the quantity of x.

So far, we have only used these variables in place of t and x, but there is a potential solution that does not need any new variables. This potential solution can be found by doing a search on *web sites* that help you *solveperfect squaretrinomials*.

## Practicing by trying some examples

Trying some examples is the best way to **learn new concepts**. Even though this section is called Try Some Examples, these examples can be done on a computer or mobile device.

This section of the article is for those who do not have a computer or mobile device. You can still try these examples!

There are several problems you can try these examples on. For example, you can **try trying trying trying trying trying trying tries** on: 3, 5, 7, and 11 times the expression x2 + x + 1 = 0. Or you can **try trying tries like adding 1** to each term and finding the solution that does not **change unless x** + 1 is removed.