To show that the expressions \((x−5)(x+3)\) and \(2x^2−x^2−2x−20+5\) are equivalent, we will use the Distributive Property on Expression 1 and simplify Expression 2.
Expression 1: \((x−5)(x+3)\)
Explanation: The expression is equivalent to the sum of the products of each term in the first binomial with each term in the second binomial.
Steps:
- Distribute \(x\) to both terms in \((x + 3)\): \[ x \cdot (x + 3) = x^2 + 3x \]
- Distribute \(-5\) to both terms in \((x + 3)\): \[ -5 \cdot (x + 3) = -5x - 15 \]
- Combine the results: \[ x^2 + 3x - 5x - 15 = x^2 - 2x - 15 \]
Expression 2: \(2x^2−x^2−2x−20+5\)
Explanation: This expression is equivalent to performing standard operations, combining like terms to simplify.
Steps:
- Combine \(2x^2\) and \(-x^2\): \[ 2x^2 - x^2 = x^2 \]
- Combine the constant terms: \[ -20 + 5 = -15 \]
- Combine all terms: \[ x^2 - 2x - 15 \]
Conclusion
Both expressions simplify to \(x^2 - 2x - 15\), thus proving that \((x−5)(x+3)\) is equivalent to \(2x^2−x^2−2x−20+5\).