An equivalent expression gives the same output for every value of . For a one-variable multiple-choice question, graph the stated expression in Desmos and compare promising choices by full-curve overlap. A shared intercept is only one matching input, not a match everywhere. When you subtract parentheses, watch the minus sign on every term.
Hints
- Hint 1
The difference of two expressions means the first expression minus the entire second expression. Keep the second one in parentheses when you type it. Which terms will the minus sign affect?
- Hint 2
An equivalent expression must give the same output for every . At , compare the original graph’s intercept with each choice’s constant term. Then check whether a remaining choice overlaps the whole curve, not just that point.
Step-by-step
Approach 1: Compare the graphs in Desmos
- Step 1
Graph the difference and check its intercept
Difference means subtract the second expression from the first. Type in Desmos. Click its -intercept, where ; Desmos shows . So choices with constant terms or already fail.
- Step 2
Test a choice with the same intercept
Type on line 2. Its graph meets the original at but separates away from that point. One shared output isn't enough: equivalent expressions agree at every .
- Step 3
Find the full-curve match
Replace line 2 with . Desmos shows full overlap throughout the window, not just an intersection. So the expression equivalent to the given difference is . Choice B.
Approach 2: Track the subtraction by hand
- Step 1
Change every sign in the second expression
Subtracting a whole expression changes the sign of every term inside its parentheses. Multiply each term in the second parentheses by :
In particular, subtracting gives .
- Step 2
Group like terms
Like terms have the same variable part, so terms combine with terms, not terms. Group them:
- Step 3
Combine the three groups
Combine the squared terms, the terms, and the constants:
So the difference is . Choice B.