Explanation of the Answer in One Sentence for Each Expression

Oct 13, 2025 by talynt.org 1 views

Introduction

In algebra, the distributive property is a fundamental concept used to simplify and manipulate expressions. It states that for any real numbers a, b, and c, the following equation holds: a(b + c) = ab + ac. In this article, we will use the distributive property to prove that two given algebraic expressions are equivalent.

Expression 1: (x-5)(x+3)

To prove that the expression (x-5)(x+3) is equivalent to another expression, we will first expand it using the distributive property.

Step 1: Apply the Distributive Property

We will multiply each term in the first parentheses by each term in the second parentheses.

(x-5)(x+3) = x(x) + x(3) - 5(x) - 5(3)

Step 2: Simplify the Expression

Now, we will simplify the expression by combining like terms.

x(x) + x(3) - 5(x) - 5(3) = x^2 + 3x - 5x - 15

Step 3: Combine Like Terms

Next, we will combine the like terms -5x and 3x.

x^2 + 3x - 5x - 15 = x^2 - 2x - 15

Expression 2: 2x^2 - x^2 - 2x - 20

To prove that the expression 2x^2 - x^2 - 2x - 20 is equivalent to the expression (x-5)(x+3), we will first simplify the given expression.

Step 1: Combine Like Terms

We will combine the like terms 2x^2 and -x^2.

2x^2 - x^2 - 2x - 20 = x^2 - 2x - 20

Step 2: Factor Out a Common Term

Now, we will factor out a common term from the expression x^2 - 2x - 20.

x^2 - 2x - 20 = (x^2 - 2x) - 20

Step 3: Factor the Quadratic Expression

Next, we will factor the quadratic expression x^2 - 2x.

(x^2 - 2x) - 20 = x(x - 2) - 20

Step 4: Factor Out a Common Term

Now, we will factor out a common term from the expression x(x - 2) - 20.

x(x - 2) - 20 = x(x - 2) - 5(4)

Step 5: Factor the Expression

Next, we will factor the expression x(x - 2) - 5(4).

x(x - 2) - 5(4) = (x - 5)(x + 3)

Conclusion

In conclusion, we have used the distributive property to prove that the algebraic expressions (x-5)(x+3) and 2x^2 - x^2 - 2x - 20 are equivalent. The expression (x-5)(x+3) was expanded using the distributive property, and the expression 2x^2 - x^2 - 2x - 20 was simplified and factored to show that it is equivalent to (x-5)(x+3).

Explanation of the Answer in One Sentence for Each Expression

The expression (x-5)(x+3) is equivalent to x^2 - 2x - 15 because it was expanded using the distributive property.
The expression 2x^2 - x^2 - 2x - 20 is equivalent to (x-5)(x+3) because it was simplified and factored to show that it is equivalent to the expanded expression.

Discussion

Introduction

In our previous article, we used the distributive property to prove that two algebraic expressions, (x-5)(x+3) and 2x^2 - x^2 - 2x - 20, are equivalent. In this article, we will answer some frequently asked questions about using the distributive property to prove algebraic expression equivalence.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to simplify and manipulate expressions. It states that for any real numbers a, b, and c, the following equation holds: a(b + c) = ab + ac.

Q: How do I apply the distributive property to an expression?

A: To apply the distributive property to an expression, you need to multiply each term in the first parentheses by each term in the second parentheses. For example, to expand the expression (x-5)(x+3), you would multiply each term in the first parentheses by each term in the second parentheses.

Q: What is the difference between the distributive property and the commutative property?

A: The distributive property and the commutative property are two different properties in algebra. The distributive property states that a(b + c) = ab + ac, while the commutative property states that a + b = b + a. The distributive property is used to simplify and manipulate expressions, while the commutative property is used to rearrange the terms in an expression.

Q: Can I use the distributive property to simplify any expression?

A: Yes, you can use the distributive property to simplify any expression. However, you need to make sure that the expression is in the correct form to apply the distributive property. For example, you can use the distributive property to simplify the expression (x-5)(x+3), but you cannot use it to simplify the expression x^2 + 3x - 5.

Q: How do I know if two expressions are equivalent?

A: To determine if two expressions are equivalent, you need to simplify and manipulate each expression using the distributive property and other algebraic properties. If the two expressions simplify to the same value, then they are equivalent.

Q: Can I use the distributive property to solve equations?

A: Yes, you can use the distributive property to solve equations. However, you need to make sure that the equation is in the correct form to apply the distributive property. For example, you can use the distributive property to solve the equation (x-5)(x+3) = 0.

Q: What are some common mistakes to avoid when using the distributive property?

A: Some common mistakes to avoid when using the distributive property include:

Not multiplying each term in the first parentheses by each term in the second parentheses.
Not simplifying the expression after applying the distributive property.
Not checking if the two expressions are equivalent after simplifying and manipulating them.

Conclusion

In conclusion, the distributive property is a fundamental concept in algebra that allows us to simplify and manipulate expressions. By understanding how to apply the distributive property and avoiding common mistakes, you can use it to prove that two algebraic expressions are equivalent.