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


say it like how a 8th grader would

Answer 1

To prove that Expression 1 and Expression 2 are equivalent, we will first simplify Expression 1 using the Distributive Property, then we’ll simplify Expression 2 and see if they match.

**Expression 1: (x−5)(x+3)**
Using the Distributive Property, we multiply the two binomials:
1. First, distribute \(x\) from the first parenthesis:
\(x \cdot x + x \cdot 3 = x^2 + 3x\)
2. Next, distribute \(-5\) from the first parenthesis:
\(-5 \cdot x - 5 \cdot 3 = -5x - 15\)
3. Now, combine all the terms together:
\(x^2 + 3x - 5x - 15\)
Which simplifies to:
\(x^2 - 2x - 15\)

**Expression 2: 2x^2 − x^2 − 2x − 20 + 5**
Now let’s simplify Expression 2 by combining like terms:
1. Combine \(2x^2\) and \(-x^2\):
\(2x^2 - x^2 = x^2\)
2. Combine the constants \(-20 + 5\):
\(-20 + 5 = -15\)
3. So now we have:
\(x^2 - 2x - 15\)

Now, comparing both simplified expressions:
- Expression 1 gives us \(x^2 - 2x - 15\)
- Expression 2 also simplifies to \(x^2 - 2x - 15\)

Since both expressions simplify to the same algebraic expression \(x^2 - 2x - 15\), we can say they are equivalent.

So, the final answer is:
**Expression 1** simplifies to **\(x^2 - 2x - 15\)** and **Expression 2** simplifies to **\(x^2 - 2x - 15\)**, proving they are equivalent!