To prove that Expression 1 and Expression 2 are equivalent, we will first simplify Expression 1 using the Distributive Property, then we’ll simplify Expression 2 and see if they match.
Expression 1: (x−5)(x+3)
Using the Distributive Property, we multiply the two binomials:
- First, distribute \(x\) from the first parenthesis:
\(x \cdot x + x \cdot 3 = x^2 + 3x\) - Next, distribute \(-5\) from the first parenthesis:
\(-5 \cdot x - 5 \cdot 3 = -5x - 15\) - Now, combine all the terms together:
\(x^2 + 3x - 5x - 15\)
Which simplifies to:
\(x^2 - 2x - 15\)
Expression 2: 2x^2 − x^2 − 2x − 20 + 5
Now let’s simplify Expression 2 by combining like terms:
- Combine \(2x^2\) and \(-x^2\):
\(2x^2 - x^2 = x^2\) - Combine the constants \(-20 + 5\):
\(-20 + 5 = -15\) - So now we have:
\(x^2 - 2x - 15\)
Now, comparing both simplified expressions:
- Expression 1 gives us \(x^2 - 2x - 15\)
- Expression 2 also simplifies to \(x^2 - 2x - 15\)
Since both expressions simplify to the same algebraic expression \(x^2 - 2x - 15\), we can say they are equivalent.
So, the final answer is:
Expression 1 simplifies to \(x^2 - 2x - 15\) and Expression 2 simplifies to \(x^2 - 2x - 15\), proving they are equivalent!
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