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.
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Expression 1: (x−5)(x+3)
Expression 2: 2x^2−x^2−2x−20+5
Expression 1: (x-5)(x+3) can be expanded using the distributive property to give x^2 + 3x - 5x - 15, which simplifies to x^2 - 2x - 15.
Expression 2: Using the commutative property for addition, we can rearrange the terms 2x^2, -x^2, -2x, -20, and 5 to give 2x^2 - x^2 - 2x - 20 + 5.
By combining like terms -x^2 and 2x^2, and combining the constants -20 and 5, we can simplify Expression 2 to x^2 - 2x - 15, which is equal to Expression 1.
To prove that the expressions are equivalent, we will use the Distributive Property on Expression 1.
Expression 1: (x - 5)(x + 3)
Using the Distributive Property, we multiply each term of the first parenthesis by each term of the second parenthesis:
x * x + x * 3 - 5 * x - 5 * 3
Simplifying, we get:
x^2 + 3x - 5x - 15
Combining like terms, we have:
x^2 - 2x - 15
So, Expression 1 is equivalent to x^2 - 2x - 15.
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Next, let's simplify Expression 2:
Expression 2: 2x^2 - x^2 - 2x - 20 + 5
Combining like terms, we have:
x^2 - 2x - 15
Both Expression 1 and Expression 2 simplify to x^2 - 2x - 15, proving that they are equivalent.