Question
in 1-3 sentences explain: Use the Distributive Property to prove that these algebraic expressions are equivalent. Explain your answer in one sentence for each expression and show your work:
Expression 1: (x−5)(x+3)
Expression 2: 2x^2−x^2−2x−20+5
Expression 1: (x−5)(x+3)
Expression 2: 2x^2−x^2−2x−20+5
Answers
GPT-4o mini
To prove that the expressions \((x−5)(x+3)\) and \(2x^2−x^2−2x−20+5\) are equivalent, we can simplify both expressions.
**For Expression 1**: Using the Distributive Property, we have:
\[
(x−5)(x+3) = x^2 + 3x - 5x - 15 = x^2 - 2x - 15.
\]
**For Expression 2**: Now simplify it:
\[
2x^2 - x^2 - 2x - 20 + 5 = (2x^2 - x^2) - 2x - 20 + 5 = x^2 - 2x - 15.
\]
Since both expressions simplify to \(x^2 - 2x - 15\), they are equivalent.
**For Expression 1**: Using the Distributive Property, we have:
\[
(x−5)(x+3) = x^2 + 3x - 5x - 15 = x^2 - 2x - 15.
\]
**For Expression 2**: Now simplify it:
\[
2x^2 - x^2 - 2x - 20 + 5 = (2x^2 - x^2) - 2x - 20 + 5 = x^2 - 2x - 15.
\]
Since both expressions simplify to \(x^2 - 2x - 15\), they are equivalent.