For equivalent expressions, two forms give the same value for every . On a multiple-choice question, try in Desmos: the terms containing vanish, making the constant values quick to compare. A match at one input alone doesn't prove equivalence, but a choice that fails this check can't be right. Watch the sign when subtracting parentheses.
Hints
- Hint 1
Equivalent expressions must agree at every input, including . Try that value in the original expression first. It removes the terms containing , so what value does Desmos give?
- Hint 2
Substituting zero into the second parentheses gives . You are subtracting three times that negative number, so the result adds to the value from the first parentheses. Now test each choice at zero.
Step-by-step
Approach 1: Test the choices at zero
- Step 1
Find the original expression's value at zero
Set in Desmos, then type the given expression with in place of . Desmos prints . At zero, the expression is ; subtracting a negative adds, so its value is .
- Step 2
Eliminate choices that disagree
Type each choice using . Desmos prints , , , and , in order. Only gives the required value of . One matching input doesn't prove two expressions are equivalent on its own, but every equivalent choice must pass this check, and only one does. Choice D.
Approach 2: Distribute to prove the match
- Step 1
Distribute the five
The distributive property means multiplying every term inside the parentheses. Multiply by both and :
Keep the minus before the second parentheses in place for the next move.
- Step 2
Distribute the negative three
The factor multiplies both terms inside the second parentheses. In particular, . Distribute it:
- Step 3
Combine matching terms
Like terms have the same variable part. Group the -terms and the numbers:
Add within each group: . This rewrite works for every , so the equivalent expression is . Choice D.