Which expression is equivalent to ?
For expression-equivalence questions, first rewrite the given expression into a simple standard form by expanding and combining like terms, using patterns such as . Once simplified, factor if needed (often by taking out a greatest common factor) and then match your result to the answer choices. This expand–simplify–factor approach is quick, reduces mistakes, and makes it easy to see which option is truly equivalent.
Hints
Start with the squared binomial
Focus first on the part . How can you expand this using the pattern for squaring a sum, ?
Simplify before you factor
After you expand , carefully subtract and combine like terms. You should end up with a simpler polynomial in standard form.
Look for a common factor
Once you have the simplified polynomial, check if all terms share a common factor that you can factor out.
Desmos Guide
Graph the original expression
In Desmos, enter the original expression as y1 = (x + 7)^2 - 49 to create its graph.
Graph each answer choice
For each answer choice, type its expression as y2, y3, etc. For example, take the expression from option A and type it exactly after y2 =, then do the same with options B, C, and D as separate functions.
Compare the graphs
Look at the graphs of y1 and each of the other functions. The choice that is equivalent will have a graph that lies exactly on top of y1 for all values of (the two graphs are the same curve everywhere). That option is the equivalent expression.
Step-by-step Explanation
Expand the squared term
Use the formula for squaring a sum: .
Here, and :
So the expression becomes
Combine like terms
Now simplify by combining the constant terms and :
So the original expression simplifies to .
Factor the simplified expression
Factor by taking out the greatest common factor, which is :
Therefore, is equivalent to .