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: 2x2−x2−2x−20+5
To prove that Expression 1 is equivalent to Expression 2 using the Distributive Property, we can expand Expression 1: (x-5)(x+3) = x(x) + x(3) -5(x) -5(3) = x^2 + 3x -5x -15 = 2x^2 -2x -15, which is equivalent to Expression 2 by combining like terms and rearranging them.
To prove that the expressions are equivalent, we need to apply the Distributive Property to Expression 1:
Expression 1: (x - 5)(x + 3)
= x(x) + x(3) - 5(x) - 5(3) (multiplying each term inside the parentheses by each term outside the parentheses)
= x^2 + 3x - 5x - 15 (simplifying multiplication)
= x^2 - 2x - 15 (combining like terms)
For Expression 2, we simplify it by combining like terms:
Expression 2: 2x^2 - x^2 - 2x - 20 + 5
= (2 - 1)x^2 - 2x - 15 (simplifying -x^2 to -1x^2)
= x^2 - 2x - 15 (combining like terms)
Both expressions simplify to x^2 - 2x - 15, proving that they are equivalent.