An equivalent expression gives the same value for every input, not just one. For a one-variable question with complete answer choices, graph the given expression in Desmos and compare a choice with it. The matching curves overlap throughout the window. When checking by hand, remember that a minus before parentheses changes every sign inside.
Hints
- Hint 1
Put the given expression after in Desmos, then graph an answer choice on another line. Equivalent expressions draw the same curve wherever you look, not merely curves that meet at one point.
- Hint 2
The -intercept is where a graph meets the vertical axis, at . Different intercepts rule out a choice, but matching intercepts aren't enough. Do the curves stay together away from that point?
Step-by-step
Approach 1: Compare the curves in Desmos
- Step 1
Graph the given expression
Type on line 1. Desmos draws a curve; click where it meets the vertical axis to see . That point gives the expression's value when .
- Step 2
Check a tempting sign choice
Type on line 2. Desmos shows that it meets line 1 at but pulls away elsewhere. One shared point doesn't make two expressions equivalent.
- Step 3
Find the full-curve match
Add on line 3. Desmos shows line 3 covering the original across the whole window, while line 2 separates from them. The equivalent expression is . Choice D.
Approach 2: See why the signs give that match
- Step 1
Subtract the whole middle expression
The distributive property says a factor multiplies every term inside parentheses. Treat the minus as , so all three signs in the middle expression change:
- Step 2
Distribute the 2
Multiply both terms in the last parentheses by , including :
- Step 3
Group terms that match
Group like terms, which have the same variable part. Keep the terms, the terms, and the numbers in separate groups:
- Step 4
Combine each group
Add within each group: .So the algebra gives the same expression that overlapped in Desmos: . Choice D.