For equivalent expressions, the matching choice must give the same value as the original for every . A quick Desmos move is to evaluate the original and each choice at one convenient input. If only one choice matches, the others are ruled out. A single matching value isn't proof by itself; distributing and combining like terms shows why an expression works for every input.
Hints
- Hint 1
Pick a small input, such as , and substitute it into the original expression. Use the same input for every answer choice so you can compare their outputs fairly.
- Hint 2
Two expressions can agree at one input without being equivalent. But a choice that gives a different output at your test input is definitely wrong. Do any choices match the original?
Step-by-step
Approach 1: Test the choices in Desmos
- Step 1
Find the original expression's output
Choose as a small test input. Type , then in Desmos. Using tells Desmos to substitute your chosen number rather than graph . It shows for the original expression.
- Step 2
Compare every choice at the same input
Type each choice below the original, replacing its with . Desmos shows , , , and , in choice order. Only matches the original's . One different output rules out an equivalent expression. The matching expression is . Choice B.
Approach 2: Distribute to confirm the match
- Step 1
Multiply through the first parentheses
Use the distributive property: the multiplies both terms inside its parentheses. Distribute :
- Step 2
Multiply through the second parentheses
The also multiplies both terms, including the . Distribute :
- Step 3
Combine the variable terms
Like terms have the same variable part, so and can be added. Combine them:
- Step 4
Combine the constants
The remaining numbers have no . Combine them, keeping the minus sign on : . So the equivalent expression is . Choice B.