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.