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 show that Expression 1 \((x - 5)(x + 3)\) is equivalent to Expression 2 \(2x^2 - x^2 - 2x - 20 + 5\) using the Distributive Property, we will simplify both expressions and see if they are equal.

**Step 1: Simplify Expression 1**

To simplify Expression 1:
\[
(x - 5)(x + 3) = x(x + 3) - 5(x + 3)
\]
Now apply the distributive property:
\[
= x^2 + 3x - 5x - 15
\]
Combine like terms:
\[
= x^2 - 2x - 15
\]

**Step 2: Simplify Expression 2**

Now, let's simplify Expression 2:
The expression is \(2x^2 - x^2 - 2x - 20 + 5\).
First, combine \(2x^2 - x^2\):
\[
= x^2 - 2x - 20 + 5
\]
Now combine the constant terms:
\[
= x^2 - 2x - 15
\]

**Conclusion:**

Both expressions simplify to:
\[
x^2 - 2x - 15
\]
Thus, \((x - 5)(x + 3)\) is equivalent to \(2x^2 - x^2 - 2x - 20 + 5\) because when fully simplified, both expressions yield the exact same result of \(x^2 - 2x - 15\).