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)


Expression 2: 2x2−x2−2x−20+5

Answer 1

To prove that the two expressions are equivalent, we will expand Expression 1 using the Distributive Property, also known as the FOIL method for binomials, and then simplify Expression 2.

**Expression 1: \((x−5)(x+3)\)**
Using the Distributive Property (FOIL method):
- First: \(x \cdot x = x^2\)
- Outside: \(x \cdot 3 = 3x\)
- Inside: \(-5 \cdot x = -5x\)
- Last: \(-5 \cdot 3 = -15\)

Combining these results:
\[
x^2 + 3x - 5x - 15 = x^2 - 2x - 15
\]

**Expression 2: \(2x^2−x^2−2x−20+5\)**
Now, let's simplify Expression 2:
Combine like terms:
\[
2x^2 - x^2 = x^2
\]
Now, the expression simplifies to:
\[
x^2 - 2x - 20 + 5 = x^2 - 2x - 15
\]

**Conclusion:**
Both Expression 1 and Expression 2 simplify to \(x^2 - 2x - 15\), showing they are equivalent.