Question
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: 2x2−x2−2x−20+5
Expression 1: (x−5)(x+3)
Expression 2: 2x2−x2−2x−20+5
Answers
Answer
turn it all into words
Answered by
GPT-4o mini
To prove that the expressions \((x−5)(x+3)\) and \(2x^2−x^2−2x−20+5\) are equivalent using the Distributive Property, we will expand Expression 1 and simplify Expression 2.
**Step 1: Expand Expression 1**
\[
(x−5)(x+3) = x(x+3) - 5(x+3)
\]
Distributing \(x\) and \(-5\):
\[
= x^2 + 3x - 5x - 15
\]
Combine like terms:
\[
= x^2 - 2x - 15
\]
**Step 2: Simplify Expression 2**
Expression 2 is given as:
\[
2x^2−x^2−2x−20+5
\]
Combine like terms:
\[
= (2x^2 - x^2) - 2x - 20 + 5
\]
This simplifies to:
\[
= x^2 - 2x - 15
\]
**Conclusion:**
Now we can see that both expressions equal \(x^2 - 2x - 15\). Thus, we have shown that the expressions \((x−5)(x+3)\) and \(2x^2−x^2−2x−20+5\) are equivalent because they simplify to the same polynomial \(x^2 - 2x - 15\).
**Step 1: Expand Expression 1**
\[
(x−5)(x+3) = x(x+3) - 5(x+3)
\]
Distributing \(x\) and \(-5\):
\[
= x^2 + 3x - 5x - 15
\]
Combine like terms:
\[
= x^2 - 2x - 15
\]
**Step 2: Simplify Expression 2**
Expression 2 is given as:
\[
2x^2−x^2−2x−20+5
\]
Combine like terms:
\[
= (2x^2 - x^2) - 2x - 20 + 5
\]
This simplifies to:
\[
= x^2 - 2x - 15
\]
**Conclusion:**
Now we can see that both expressions equal \(x^2 - 2x - 15\). Thus, we have shown that the expressions \((x−5)(x+3)\) and \(2x^2−x^2−2x−20+5\) are equivalent because they simplify to the same polynomial \(x^2 - 2x - 15\).
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