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: 2x^2−x^2−2x−20+5

1 answer

To prove that the expressions \((x−5)(x+3)\) and \(2x^2−x^2−2x−20+5\) are equivalent using the Distributive Property, we will simplify both expressions and compare them.

Step 1: Simplify Expression 1: \((x−5)(x+3)\)

Using the Distributive Property (or the FOIL method for binomials):

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

Step 2: Simplifying Expression 2: \(2x^2−x^2−2x−20+5\)

Combine like terms:

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

Final Comparison:

Now that we have both expressions simplified, we see:

Expression 1: \(x^2 - 2x - 15\)
Expression 2: \(x^2 - 2x - 15\)

Both expressions are identical, thus they are equivalent.

In summary: The first expression simplifies to \(x^2 - 2x - 15\) by applying the Distributive Property. The second expression simplifies to the same result, confirming that both expressions are equivalent.