To prove that the two algebraic expressions (x−5)(x+3) and 2x2−x2−2x−20+5 are equivalent using the Distributive Property, we'll expand Expression 1 and simplify Expression 2.

Question

**Expression 1: $(x-5)(x+3)$**
Using the Distributive Property (also known as the FOIL method for binomials), we distribute each term in the first binomial by each term in the second binomial:

$$(x-5)(x+3)=x\cdot x+x\cdot 3-5\cdot x-5\cdot 3$$
$$=x^2+3x-5x-15$$
$$=x^2-2x-15$$

**Expression 2: $2x^2-x^2-2x-20+5$**
Now, we simplify Expression 2 by combining like terms:

$$2x^2-x^2=(2-1)x^2=x^2$$
$$-2x$$
$$-20+5=-15$$
Thus, combining everything, we have:
$$Expression2=x^2-2x-15$$

Now, since both Expression 1 and Expression 2 simplify to $x^2-2x-15$, we conclude that the two expressions are equivalent.

**Summary:**
Expression 1 simplifies to $x^2-2x-15$ via the Distributive Property. Expression 2 simplifies to $x^2-2x-15$ by combining like terms; therefore, both expressions are equivalent.
make tis smaller

Answer 1

To prove that the expressions $(x-5)(x+3)$ and $2x^2-x^2-2x-20+5$ are equivalent, we'll expand Expression 1 and simplify Expression 2.

Expression 1: $(x-5)(x+3)$ Using the Distributive Property (FOIL method):

$$(x-5)(x+3)=x\cdot x+x\cdot 3-5\cdot x-5\cdot 3$$ $$=x^2+3x-5x-15$$ $$=x^2-2x-15$$

Expression 2: $2x^2-x^2-2x-20+5$ Now, simplify Expression 2 by combining like terms:

$$2x^2-x^2=x^2$$ $$-2x$$ $$-20+5=-15$$ Thus, we have: $$\text{Expression 2}=x^2-2x-15$$

Since both expressions simplify to $x^2-2x-15$, we conclude that they are equivalent.