To prove that the two expressions are equivalent using the Distributive Property, we need to simplify both expressions and see if they yield the same result.
Expression 1: (x−5)(x+3)
Using the Distributive Property (also known as the FOIL method for binomials), we multiply each term in the first parentheses by each term in the second parentheses:
\[ (x-5)(x+3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3 \]
Calculating this step-by-step:
- \( x \cdot x = x^2 \)
- \( x \cdot 3 = 3x \)
- \( -5 \cdot x = -5x \)
- \( -5 \cdot 3 = -15 \)
Now, combine these results:
\[ x^2 + 3x - 5x - 15 = x^2 - 2x - 15 \]
Expression 2: 2x^2−x^2−2x−20+5
Now, we'll simplify Expression 2 by combining like terms:
-
Combine \( 2x^2 \) and \( -x^2 \): \[ 2x^2 - x^2 = x^2 \]
-
Next, combine the linear terms and the constants: \[ -2x \] \[ -20 + 5 = -15 \]
Putting everything together, we have:
\[ x^2 - 2x - 15 \]
Conclusion
Now we compare both expressions after simplification:
- Expression 1 simplified to: \( x^2 - 2x - 15 \)
- Expression 2 simplified to: \( x^2 - 2x - 15 \)
Since both expressions are equal, we conclude that:
Expression 1, \((x−5)(x+3)\), simplifies to \(x^2 - 2x - 15\) and Expression 2, \(2x^2−x^2−2x−20+5\), also simplifies to \(x^2 - 2x - 15\), thus proving they are equivalent.
3 answers