Which of the following is equivalent to ?
For expressions like this, first check whether both terms share a common factor (such as the same parentheses). If they do, factor that piece out and combine the numerical coefficients in front, which is often faster and less error-prone than distributing immediately. Then, if needed, distribute once at the end to write the expression in standard form . If factoring isn’t obvious, distribute carefully and combine like terms, watching signs closely.
Hints
Look for a shared factor
Both terms in contain the same parentheses. How can you use that to combine them into one term?
Combine the numerical coefficients
If you factor out , what operation do you perform on the numbers and in front of it?
Simplify completely
Once you have something like , distribute to both and to get a final form .
Desmos Guide
Enter the original expression
In Desmos, type f(x) = 5(x - 3) - 2(x - 3) on the first line. This represents the expression in the question.
Enter each answer choice as a separate function
On the next lines, enter:
g(x) = 3x + 9h(x) = 7x - 6p(x) = 3(x + 3)q(x) = 3x - 9
These represent the four choices.
Compare graphs or table values
Look at the graphs: the correct choice will have a line that exactly overlaps the graph of for all . You can also create a table for each function (click the gear icon and "Table") and compare to each other function for several -values; the equivalent expression will match at every point.
Step-by-step Explanation
Notice the common factor
Both parts of the expression share the factor :
Because appears in both terms, you can factor it out, just like you would factor out a common number.
Factor out and combine coefficients
Write the expression with factored out:
Now combine the coefficients:
So the original expression simplifies to a single product involving and .
Distribute to get the final simplified form
Now expand using the distributive property:
So is equivalent to , which matches choice D.