Which of the following is equivalent to ?
For equivalent-expression questions, first look for common factors or repeating pieces; here, both terms share , so factor that out instead of fully expanding. If factoring is not obvious, you can expand, combine like terms carefully, and then factor or compare to the answer choices’ expanded forms—whichever path is shorter and less error-prone. Always check your work by quickly re-expanding your factored form to be sure it matches the original expression.
Hints
Look for a common factor
In , both terms involve the same binomial. What is the repeated binomial factor you see in each term?
Rewrite to make the common factor obvious
Try rewriting as so that both terms clearly show the same factor. Then think about how to factor an expression when each term shares that factor.
Finish the factoring
After you factor out the common binomial, simplify what remains inside the brackets, and then compare that product with the answer choices.
Desmos Guide
Graph the original expression
In Desmos, enter y1 = (x-3)^2 + 2(x-3) to graph the original expression. This will be your reference curve.
Graph each answer choice for comparison
Enter each option as a separate function, for example y2 = x^2 - 9, y3 = x^2 - 6x + 5, y4 = (x-1)^2, and y5 = (x-3)(x-1) (or whichever labeling you prefer).
Decide which expression is equivalent
Look at the graphs: the correct choice will be the one whose graph lies exactly on top of the graph of y1 for all visible -values (they coincide everywhere). That option is the expression equivalent to .
Step-by-step Explanation
Factor out the common binomial
Start with the expression
Rewrite as so you can see the common factor more clearly:
Both terms contain the factor , so factor it out:
Simplify the remaining factor and match the choice
Now simplify the binomial inside the brackets:
So the whole expression becomes
Therefore, the expression is equivalent to .