Question
Use the Distributive Property to show that these algebraic expressions are equivalent.
Expression 1: (x−5)(x+3)
Expression 2: 2x2−x2−2x−20+5
Explain your answer in one sentence for each expression (2 pts)
Show the steps of your work for each expression (2 pts)
(make it sound like a high schooler)
Expression 1: (x−5)(x+3)
Expression 2: 2x2−x2−2x−20+5
Explain your answer in one sentence for each expression (2 pts)
Show the steps of your work for each expression (2 pts)
(make it sound like a high schooler)
Answers
Answered by
GPT-4o mini
To show that the two expressions \((x - 5)(x + 3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent, we’ll first simplify each expression using the Distributive Property and then compare the results.
### Expression 1:
**Step 1: Apply the Distributive Property**
We’ll distribute \((x - 5)\) with \((x + 3)\):
\[
(x - 5)(x + 3) = x(x + 3) - 5(x + 3)
\]
**Step 2: Simplify**
Distributing each part:
- \(x(x + 3) = x^2 + 3x\)
- \(-5(x + 3) = -5x - 15\)
Now combine those:
\[
x^2 + 3x - 5x - 15 = x^2 - 2x - 15
\]
### Expression 2:
**Step 1: Combine like terms**
Starting with \(2x^2 - x^2 - 2x - 20 + 5\):
- Combine \(2x^2 - x^2 = x^2\)
- Combine \(-20 + 5 = -15\)
So, the expression simplifies to:
\[
x^2 - 2x - 15
\]
### Conclusion:
Both expressions simplify to \(x^2 - 2x - 15\), proving they are equivalent.
In Expression 1, we showed how distributing the terms results in \(x^2 - 2x - 15\) through the Distributive Property. In Expression 2, we combined like terms to arrive at the same result, demonstrating the equivalence of both expressions.
### Expression 1:
**Step 1: Apply the Distributive Property**
We’ll distribute \((x - 5)\) with \((x + 3)\):
\[
(x - 5)(x + 3) = x(x + 3) - 5(x + 3)
\]
**Step 2: Simplify**
Distributing each part:
- \(x(x + 3) = x^2 + 3x\)
- \(-5(x + 3) = -5x - 15\)
Now combine those:
\[
x^2 + 3x - 5x - 15 = x^2 - 2x - 15
\]
### Expression 2:
**Step 1: Combine like terms**
Starting with \(2x^2 - x^2 - 2x - 20 + 5\):
- Combine \(2x^2 - x^2 = x^2\)
- Combine \(-20 + 5 = -15\)
So, the expression simplifies to:
\[
x^2 - 2x - 15
\]
### Conclusion:
Both expressions simplify to \(x^2 - 2x - 15\), proving they are equivalent.
In Expression 1, we showed how distributing the terms results in \(x^2 - 2x - 15\) through the Distributive Property. In Expression 2, we combined like terms to arrive at the same result, demonstrating the equivalence of both expressions.