Creating equivalent forms of an expression is a high emphasis topic on the TASC Test Assessing Secondary Completion™ Mathematics subtest. Expressions, like people, are multifaceted. You can't know someone's life story by simply glancing at them. You can’t understand what expressions are representing by a quick look; they require you to work through them for understanding.

*A mathematical expression can be considered a mathematical thought or idea communicated by the language of mathematics.* Find a good review of interpreting expressions that represent a quantity in a previous TASC test math blog post.

You may be asked to **choose and produce an equivalent form of an expression to explain the properties of the quantity represented by the expression**.

Let’s look at a few examples from Khan Academy:

**Example 1:** Which expressions are equivalent to 2(4f+2g)?

- 8
*f*+ 4*g* - 2
*f*(4+2*g*) - 8
*f*+ 2*g* - 4 (2
*f*+*g*)

The goal of this example is to manipulate 2(4*f*+2*g*) in order to get one more of the answers.

First, distribute the two. By doing this you get:

2·4*f* + 2·2*g* = 8*f* + 4*g*

So we know that (a) is **an equivalent form of the original expression.** But what about the other options? By solving the same way, we can distribute the two:

b. 2*f* (4+2*g*) = 8*f* + 4*fg* (not an equivalent form of the original expression)

- 8
*f*+ 2*g*= 10*fg*(not an equivalent form of the original expression) - 4 (2
*f*+*g*) = 8*f*+ 4*g***(an equivalent form of the original expression)**

**Example 2:** Find an equivalent expression to 6*l* + 5*m* - 3*n*.

- 3 (2
*l*–*n*) + 5*m*à 6*l*– 3*n*+ 5*m*=**equivalent form**. - 3
*n*+ 6*l*– 5*m*à This is**not equivalent**to the original. The 3n is negative and 5m is positive in the original, and in this form, the 3 is positive and the 5 is now negative. - 5
*m*+ (6*l*– 3*n*) à You can remove the parenthesis and it will not change the outcome. It is an**equivalent form**.

The equivalent expressions are (a) and (c).

Writing a function that’s defined by an expression in different but equivalent forms is another important skill that the TASC Mathematics subtest will test you on.

**Example 1:** The quantity of a car C after time *t* can be written *C*(*t*) = 1.02* ^{t}* . Is the quantity of the car growing or decaying? Identify the percent rate of change from the equation for

*C*(

*t*).

**Answer:** C is growing at a rate of 2% for each time interval *t*.

**Example 2:** Solve *x*^{2} − 5*x* + 6 = 0.

**Answer:** *x* = 3 and *x* = 2.

Find more example function problems at Shmoop.com.