For equivalent expressions, both forms must give the same output for every . In a polynomial multiple-choice question, check first: every term containing disappears, so the graph’s -intercept lets you compare constants. Then graph the remaining candidate against the original in Desmos. Meeting at one point isn’t enough; the curves should overlap throughout the window.
Hints
- Hint 1
The constant term is what remains when , because both and become zero. Where does the graph show the expression’s value at ?
- Hint 2
The -intercept is where a graph crosses the vertical axis, so its -coordinate is . After you use that value to rule out choices, graph the remaining candidate and check more than one point.
Step-by-step
Approach 1: Check the intercept, then compare curves
- Step 1
Read the value at zero
Graph the given expression by typing . Click where it crosses the vertical axis. Desmos shows . That point means the expression equals when .
- Step 2
Verify the only possible match
At , only a choice’s constant term remains. Only one choice has constant , so type on a second line. Its curve overlaps the original across the window. A match at one point rules choices out; full-curve overlap checks the remaining choice. The equivalent form is . Choice D.
Approach 2: Expand the square and track the minus
- Step 1
Keep the middle term of the square
A square of a sum multiplies each part by both parts, so it has two cross-products. Write those products before calculating:
- Step 2
Calculate the three coefficients
Type , , and in Desmos. It prints , , and . Substitute them, keeping the positive middle term:
- Step 3
Subtract both terms in parentheses
The minus sign applies to both and . Distribute it through the parentheses:
- Step 4
Combine matching terms
Combine like terms, which have the same power of :
Evaluate the two subtractions:
That is the equivalent form. Choice D.