Question 29·Medium·Equivalent Expressions
Which of the following expressions is equivalent to the expression above?
For equivalent-expression questions that involve subtraction of similar-looking terms, first look for a common factor (often a repeated binomial like ). Use the distributive property in reverse to factor that common term instead of expanding everything. Then simplify the remaining parentheses carefully, watching the signs, and match the resulting factored form directly to one of the answer choices without doing unnecessary extra algebra.
Hints
Look for a repeated binomial
Compare the two terms in and see if they share a common factor.
Use the distributive property in reverse
If you have an expression like , you can factor it as . Try to put your expression into that form.
Be careful with the minus sign
When you factor out the common binomial, pay close attention to whether you are subtracting or adding inside the parentheses.
Desmos Guide
Enter the original expression
Type y1 = (2x - 3)(x + 5) - (x + 5) into Desmos so that it graphs the original expression.
Graph each answer choice
On new lines, enter each option as a separate function, for example y2 = (x+5)(2x-3), y3 = (2x-4)(x-5), and y4 = (x+5)(2x-2), and then the remaining choice as y5 = (x+5)(2x-4).
Compare the graphs
Look at the graphs of all the functions. The correct answer is the option whose graph lies exactly on top of the graph of y1 for all x-values (they should be indistinguishable).
Step-by-step Explanation
Notice the common factor
Rewrite the first product to highlight the common factor:
Now you can clearly see that both terms contain the factor .
Factor out the common factor
Use the distributive property in reverse: if you have , you can factor it as .
Here, , so
Simplify inside the parentheses and match the choice
Now simplify the expression inside the brackets:
So the entire expression becomes
Therefore, an equivalent expression is , which matches answer choice D.