Use the distributive property to prove that these algebraic expressions are equivalent

Question

(X-5)(x+3)
2x^2 - x^2-2x-20+5

Answer 1

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.