Question

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)
(
𝑥
−
5
)
(
𝑥
+
3
)


Expression 2: 2x2−x2−2x−20+5
make it sound like a 9th grader wrote it.

Answer 1

To prove that the two expressions \((x-5)(x+3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent, we can use the Distributive Property on Expression 1 and simplify Expression 2.

**For Expression 1:**
Let's expand \((x - 5)(x + 3)\) using the Distributive Property:
1. First, distribute \(x\) to both terms in the second parentheses:
\(x \cdot x + x \cdot 3 = x^2 + 3x\).
2. Then, distribute \(-5\) to both terms in the second parentheses:
\(-5 \cdot x - 5 \cdot 3 = -5x - 15\).
3. Now, combine all the results:
\(x^2 + 3x - 5x - 15 = x^2 - 2x - 15\).

So Expression 1 simplifies to \(x^2 - 2x - 15\).

**For Expression 2:**
Now let's simplify \(2x^2 - x^2 - 2x - 20 + 5\):
1. Combine \(2x^2\) and \(-x^2\):
\(2x^2 - x^2 = x^2\).
2. Next, combine \(-20 + 5 = -15\) to simplify it further:
We end up with \(x^2 - 2x - 15\) here as well.

So after simplifying Expression 2, we again get \(x^2 - 2x - 15\).

**Conclusion:**
Both expressions simplify to \(x^2 - 2x - 15\), which shows they are equivalent. This means \((x - 5)(x + 3)\) is the same as \(2x^2 - x^2 - 2x - 20 + 5\).