Simplify The Expression: $-5x^2 - 10x$

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us to manipulate and solve equations more efficiently. In this article, we will focus on simplifying the given expression: $-5x^2 - 10x$. We will use various techniques to rewrite the expression in a simpler form.

Understanding the Expression

The given expression is a quadratic expression, which is a polynomial of degree two. It consists of two terms: $-5x^2$ and $-10x$. The first term is a quadratic term, while the second term is a linear term.

Factoring Out the Greatest Common Factor (GCF)

One way to simplify the expression is to factor out the greatest common factor (GCF) of the two terms. The GCF of $-5x^2$ and $-10x$ is $-5x$. We can factor out $-5x$ from both terms:

$$-5x^2 - 10x = -5x(x + 2)$$

Simplifying the Expression

Now that we have factored out the GCF, we can simplify the expression further. We can rewrite the expression as:

$$-5x(x + 2) = -5x^2 - 10x$$

This is the same expression we started with, but it is now written in a simpler form.

Using the Distributive Property

Another way to simplify the expression is to use the distributive property. The distributive property states that for any real numbers $a$, $b$, and $c$:

$$a(b + c) = ab + ac$$

We can use this property to rewrite the expression:

$$-5x(x + 2) = -5x \cdot x - 5x \cdot 2$$

$$= -5x^2 - 10x$$

Simplifying the Expression Using Algebraic Manipulation

We can also simplify the expression using algebraic manipulation. We can rewrite the expression as:

$$-5x^2 - 10x = -5x(x + 2)$$

We can then factor out the GCF of $-5x$ from both terms:

$$-5x(x + 2) = -5x \cdot x - 5x \cdot 2$$

$$= -5x^2 - 10x$$

Conclusion

In this article, we have simplified the expression $-5x^2 - 10x$ using various techniques. We have factored out the greatest common factor (GCF) of the two terms, used the distributive property, and simplified the expression using algebraic manipulation. The simplified expression is $-5x(x + 2)$.

Final Answer

The final answer is: $\boxed{-5x(x + 2)}$

Additional Tips and Tricks

• When simplifying expressions, it is often helpful to look for common factors or patterns.
• The distributive property can be used to rewrite expressions in a simpler form.
• Algebraic manipulation can be used to simplify expressions and solve equations.

Common Mistakes to Avoid

• When simplifying expressions, it is easy to make mistakes by forgetting to factor out common factors or by using the distributive property incorrectly.
• It is also easy to make mistakes by not checking the final answer carefully.

Real-World Applications

Simplifying expressions is an important skill that has many real-world applications. For example, in physics, we often need to simplify expressions to solve problems involving motion and energy. In engineering, we often need to simplify expressions to design and analyze complex systems.

Conclusion

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Introduction

In our previous article, we simplified the expression $-5x^2 - 10x$ using various techniques. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions.

Q: What is the greatest common factor (GCF) of two terms?

A: The greatest common factor (GCF) of two terms is the largest expression that divides both terms without leaving a remainder. In the case of the expression $-5x^2 - 10x$, the GCF is $-5x$.

Q: How do I factor out the GCF from two terms?

A: To factor out the GCF from two terms, you need to divide both terms by the GCF. In the case of the expression $-5x^2 - 10x$, you can factor out $-5x$ by dividing both terms by $-5x$:

$$-5x^2 - 10x = -5x(x + 2)$$

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any real numbers $a$, $b$, and $c$:

$$a(b + c) = ab + ac$$

This property can be used to rewrite expressions in a simpler form.

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

A: To use the distributive property to simplify an expression, you need to multiply the expression by the distributive property. In the case of the expression $-5x(x + 2)$, you can use the distributive property to rewrite it as:

$$-5x(x + 2) = -5x \cdot x - 5x \cdot 2$$

$$= -5x^2 - 10x$$

Q: What is algebraic manipulation?

A: Algebraic manipulation is the process of rewriting an expression in a simpler form using various mathematical operations such as addition, subtraction, multiplication, and division.

Q: How do I use algebraic manipulation to simplify an expression?

A: To use algebraic manipulation to simplify an expression, you need to rewrite the expression in a simpler form using various mathematical operations. In the case of the expression $-5x^2 - 10x$, you can use algebraic manipulation to rewrite it as:

$$-5x^2 - 10x = -5x(x + 2)$$

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Forgetting to factor out common factors
• Using the distributive property incorrectly
• Not checking the final answer carefully

Q: What are some real-world applications of simplifying expressions?

A: Some real-world applications of simplifying expressions include:

• Solving problems involving motion and energy in physics
• Designing and analyzing complex systems in engineering
• Simplifying complex mathematical expressions in computer science

Conclusion

In conclusion, simplifying expressions is an important skill that has many real-world applications. By using various techniques such as factoring out the greatest common factor (GCF), using the distributive property, and simplifying the expression using algebraic manipulation, we can simplify expressions and solve equations more efficiently.