Question 82·Easy·Equivalent Expressions
Which of the following expressions is equivalent to ?
For expressions like , first look for the repeated binomial factor and factor it out, turning the expression into . This is usually faster and less error-prone than distributing both terms, though you can always double-check by distributing and then carefully combining like terms, especially the constant terms with negatives.
Hints
Look for something repeated
Compare the two terms and . What piece appears in both of them?
Use factoring instead of distributing
Once you see the common part, try factoring it out so you can combine the remaining numbers ( and ).
Alternative method: distribute, then combine
If factoring feels hard, distribute and into separately, then combine like terms carefully.
Desmos Guide
Enter the original expression
In Desmos, type f(x) = 3(2x - 5) + 4(2x - 5) to represent the original expression as a function.
Enter each answer choice as a separate function
On new lines, type g(x) = 6x - 15, h(x) = 12x - 35, p(x) = 14x - 25, and q(x) = 7(2x - 5) so each option is graphed as its own function.
Compare graphs or table values
Either (a) look at the graph and see which option’s graph lies exactly on top of for all , or (b) create a table for each function and check several -values. The option whose outputs always match is the equivalent expression.
Step-by-step Explanation
Notice the common factor
Look at and . Both terms contain the same binomial factor , just with different coefficients ( and ).
Factor out the common binomial
When two terms share a common factor, you can factor it out:
.
Now the expression is written as one coefficient multiplied by the binomial .
Simplify the coefficient and match the choice
Add the coefficients: , so becomes .
Among the answer choices, this matches choice D, so is the equivalent expression.