An equivalent expression gives the same output for every . With one variable and complete answer choices, graph the given expression in Desmos, then test the choices against it. Matching curves overlap throughout the graph; meeting at one point isn't enough. A shared intercept can make a wrong choice look promising, so check the whole curve.
Hints
- Hint 1
To check whether two expressions are equivalent, compare their outputs across the graph. Two curves that meet at one point may still have different values elsewhere. Look for full overlap.
- Hint 2
Keep the given expression on Desmos line 1. Put an answer choice on line 2, then replace it with the next choice. Which curve stays on top of the original rather than pulling away?
Step-by-step
Approach 1: Compare curves in Desmos
- Step 1
Graph the given expression
Type in Desmos. The makes Desmos draw the output for each . Keep this original curve on screen to compare with the choices.
- Step 2
Find the curve that overlaps
On line 2, type each choice after , replacing that line each time. The other three curves pull away from the original. When you type , Desmos shows full overlap. Don't settle for a shared point at : every choice passes through it. The equivalent expression is . Choice B.
Approach 2: See why the expressions match
- Step 1
Multiply by
The distributive property means multiplying each term inside the first parentheses. Start with the from :
- Step 2
Multiply by
Now multiply both terms by the from ; the last product is positive because two negatives multiply to a positive:
- Step 3
Put the products together
Add the results from multiplying by and by :
- Step 4
Combine the matching terms
Combine the like terms, which have the same power of . Only the two terms combine:
So the equivalent expression is . Choice B.