To prove that the two algebraic expressions are equivalent using the Distributive Property, we will first expand Expression 1 and then simplify Expression 2.
Expression 1: (x - 5)(x + 3)
Using the Distributive Property (also known as the FOIL method for binomials), we calculate:
\[ (x - 5)(x + 3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3 \] \[ = x^2 + 3x - 5x - 15 \] \[ = x^2 - 2x - 15 \]
Expression 2: 2x^2 - x^2 - 2x - 20 + 5
Now we simplify Expression 2 step by step:
\[ 2x^2 - x^2 - 2x - 20 + 5 \] First, combine like terms:
\[ = (2x^2 - x^2) - 2x - (20 - 5) \] \[ = x^2 - 2x - 15 \]
Now that both expressions have been simplified, we can see:
- Expression 1 simplifies to \(x^2 - 2x - 15\)
- Expression 2 simplifies to \(x^2 - 2x - 15\)
Conclusion: Since both expressions simplify to the same form \(x^2 - 2x - 15\), we have proved that the expressions \((x - 5)(x + 3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent.
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