To prove that the expressions \((x - 5)(x + 3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent, we'll simplify both expressions and show that they yield the same result.
Step 1: Simplify Expression 1 \((x - 5)(x + 3)\)
Using the Distributive Property (also known as the FOIL method for binomials): \[ (x - 5)(x + 3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3 \] Calculating each term: \[ = x^2 + 3x - 5x - 15 \] Combining like terms: \[ = x^2 - 2x - 15 \]
Step 2: Simplify Expression 2 \(2x^2 - x^2 - 2x - 20 + 5\)
Combine like terms: \[ = (2x^2 - x^2) - 2x - 20 + 5 \] Calculating the first part: \[ = x^2 - 2x - 20 + 5 \] Then combining the constants: \[ = x^2 - 2x - 15 \]
Conclusion:
Both expressions simplify to \(x^2 - 2x - 15\). Therefore, we conclude that:
- Expression 1, \((x - 5)(x + 3)\), simplifies to \(x^2 - 2x - 15\) using the Distributive Property.
- Expression 2, \(2x^2 - x^2 - 2x - 20 + 5\), also simplifies to \(x^2 - 2x - 15\) upon combining like terms.
Since both expressions are equal, we have proved that they are equivalent.
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