Use the distributive property to prove that these algebraic expressions are equivalent
(X-5)(x+3)
2x^2 - x^2-2x-20+5
To prove that the expressions are equivalent, we need to simplify both sides using the distributive property.
Starting with the left side: (X-5)(x+3)
Using the distributive property, we can multiply X by x and X by 3, then -5 by x and -5 by 3:
(X-5)(x+3) = X*x + X*3 - 5*x - 5*3
Simplifying:
(X-5)(x+3) = X^2 + 3X - 5x - 15
Now, let's simplify the right side: 2x^2 - x^2 - 2x - 20 + 5
Combine like terms:
2x^2 - x^2 = x^2
-2x + 5 = -2x + 5 (no like terms to combine)
Simplifying:
2x^2 - x^2 - 2x - 20 + 5 = x^2 - 2x - 15
Comparing the simplified expressions, we can see that:
(X-5)(x+3) = X^2 + 3X - 5x - 15
and
2x^2 - x^2 - 2x - 20 + 5 = x^2 - 2x - 15
Thus, using the distributive property, we have shown that the given algebraic expressions are equivalent.
To prove the equivalence of the expressions (X-5)(x+3) and 2x^2 - x^2 - 2x - 20 + 5 using the distributive property, let's expand the first expression by distributing X to both terms inside the second parentheses:
(X-5)(x+3) = X(x) + X(3) - 5(x) - 5(3).
This simplifies to:
X^2 + 3X - 5x - 15.
Now, let's simplify the expression 2x^2 - x^2 - 2x - 20 + 5:
2x^2 - x^2 - 2x - 20 + 5 = x^2 - 2x - 15.
As we can see, by applying the distributive property, we obtained the same expression: X^2 + 3X - 5x - 15 = x^2 - 2x - 15. Therefore, we have proved that these two algebraic expressions are equivalent.
To prove that the algebraic expressions (X-5)(x+3) and 2x^2 - x^2 - 2x - 20 + 5 are equivalent using the distributive property, we need to expand the first expression and simplify it until we obtain the second expression. Here's how to do it step by step:
Step 1: Use the distributive property to multiply the binomials (X-5) and (x+3):
(X-5)(x+3) = X(x) + X(3) - 5(x) - 5(3)
= X^2 + 3X - 5x - 15
Step 2: Simplify the equation obtained in step 1:
X^2 + 3X - 5x - 15
Step 3: Combine like terms by combining the X terms and the x terms:
X^2 - 5x + 3X - 15
Step 4: Rearrange the terms to write it in descending order of exponents:
X^2 - 5x + 3X - 15
Comparing the result of step 4, we obtained the equivalent expression as 2x^2 - x^2 - 2x - 20 + 5. Therefore, we have proven that the expressions (X-5)(x+3) and 2x^2 - x^2 - 2x - 20 + 5 are equivalent using the distributive property.