Using the distributive property, prove that these algebraic expressions are equivalent. In 1 sentence, explain your result.

Question

Expression one: (x - 5) (x + 3)
Expression two: 2x^2 - x^2 - 2x - 20 + 5
Make it short

Answer 1

To prove the equivalence, we expand Expression one:

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

Next, simplify Expression two:

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

Since both expressions simplify to \(x^2 - 2x - 15\), they are equivalent; hence, the result shows that both expressions represent the same quadratic function.