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

GPT-4o mini · Answer

To prove that the expressions are equivalent using the Distributive Property, we will expand Expression 1 and simplify Expression 2.

Expression 1: \((x - 5)(x + 3)\)

Applying the distributive property (also known as FOIL for binomials):

\[ (x - 5)(x + 3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3 \] \[ = x^2 + 3x - 5x - 15 \] \[ = x^2 - 2x - 15 \]

Expression 2: \(2x^2 - x^2 - 2x - 20 + 5\)

First, we can simplify Expression 2 by combining like terms:

\[ 2x^2 - x^2 - 2x - 20 + 5 = (2x^2 - x^2) - 2x + (5 - 20) \] \[ = x^2 - 2x - 15 \]

Now that both expressions have been simplified:

Expression 1 simplifies to \(x^2 - 2x - 15\).
Expression 2 simplifies to \(x^2 - 2x - 15\).

Since both expressions are equal, we conclude that the two algebraic expressions are equivalent.

In summary: Expression 1 expands to \(x^2 - 2x - 15\) through the Distributive Property, and Expression 2 simplifies to the same result, confirming their equivalence.