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: 2x^2−x^2−2x−20+5

Answer 1

To prove that the two algebraic expressions are equivalent using the Distributive Property, we will first expand Expression 1 and then simplify Expression 2.

**Expression 1: (x - 5)(x + 3)**
Using the Distributive Property (also known as the FOIL method for binomials), we calculate:

\[
(x - 5)(x + 3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3
\]
\[
= x^2 + 3x - 5x - 15
\]
\[
= x^2 - 2x - 15
\]

**Expression 2: 2x^2 - x^2 - 2x - 20 + 5**
Now we simplify Expression 2 step by step:

\[
2x^2 - x^2 - 2x - 20 + 5
\]
First, combine like terms:

\[
= (2x^2 - x^2) - 2x - (20 - 5)
\]
\[
= x^2 - 2x - 15
\]

Now that both expressions have been simplified, we can see:

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

**Conclusion:** Since both expressions simplify to the same form \(x^2 - 2x - 15\), we have proved that the expressions \((x - 5)(x + 3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent.