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.

Question

Expression 1: (x−5)(x+3)


Expression 2: 2x2−x2−2x−20+5


say it like how a 8th grader would and easy enough to copy and paste

Answer 1

To prove that the two expressions are equivalent using the Distributive Property, let's first simplify each expression step by step.

Expression 1: (x - 5)(x + 3)
Using the Distributive Property (also called the FOIL method for binomials), we can expand this expression:

1. First, multiply the first terms: x * x = x²
2. Then, the outer terms: x * 3 = 3x
3. Next, the inner terms: -5 * x = -5x
4. Finally, the last terms: -5 * 3 = -15

Now, combine these results:
x² + 3x - 5x - 15
Combine like terms (3x - 5x):
x² - 2x - 15

So, Expression 1 simplifies to:
x² - 2x - 15

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Expression 2: 2x² - x² - 2x - 20 + 5
Let's simplify this expression step by step:

1. First, combine the like terms (2x² - x²):
   2x² - x² = x²

Now, let's rewrite it:
x² - 2x - 20 + 5

2. Next, combine the constant terms (-20 + 5):
   -20 + 5 = -15

So, the expression simplifies to:
x² - 2x - 15

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Conclusion:
Both expressions simplify to x² - 2x - 15, which shows they are equivalent.