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-7)(x+4)
Expression 2: x^2- 3x - 28

Answer 1

To prove that the expressions \((x-7)(x+4)\) and \(x^2 - 3x - 28\) are equivalent using the Distributive Property, we will expand the first expression and compare it to the second expression.

Step 1: Expand Expression 1 using Distributive Property

\[ (x-7)(x+4) = x(x+4) - 7(x+4) \] \[ = x^2 + 4x - 7x - 28 \] \[ = x^2 - 3x - 28 \]

Step 2: Compare with Expression 2

The result from expanding Expression 1 is \(x^2 - 3x - 28\), which is identical to Expression 2.

Conclusion: We have shown that \((x-7)(x+4)\) simplifies to \(x^2 - 3x - 28\), hence the two expressions are equivalent.