Factor The Quadratic Expression: $x^2 + 2x - 35$A. $x - 1$ B. $x + 7$ C. $x + 1$ D. $x - 7$

Introduction

In algebra, factoring quadratic expressions is a crucial skill that helps us simplify complex equations and solve problems more efficiently. A quadratic expression is a polynomial of degree two, which means it has a highest power of two. Factoring a quadratic expression involves expressing it as a product of two binomials. In this article, we will focus on factoring the quadratic expression $x^2 + 2x - 35$.

What is Factoring?

Factoring is the process of expressing an algebraic expression as a product of simpler expressions. In the case of quadratic expressions, we can factor them into two binomials. For example, the quadratic expression $x^2 + 5x + 6$ can be factored as $(x + 2)(x + 3)$.

The Quadratic Expression $x^2 + 2x - 35$

The quadratic expression $x^2 + 2x - 35$ is a polynomial of degree two. To factor it, we need to find two numbers whose product is $-35$ and whose sum is $2$. These numbers are $7$ and $-5$, since $7 \times (-5) = -35$ and $7 + (-5) = 2$.

Factoring the Quadratic Expression

Using the numbers $7$ and $-5$, we can write the quadratic expression $x^2 + 2x - 35$ as:

$x^2 + 7x - 5x - 35$

Now, we can factor by grouping:

$(x^2 + 7x) + (-5x - 35)$

Factoring out the common terms, we get:

$x(x + 7) - 5(x + 7)$

Now, we can see that both terms have a common factor of $(x + 7)$. Factoring this out, we get:

$(x - 5)(x + 7)$

Conclusion

In this article, we have factored the quadratic expression $x^2 + 2x - 35$ using the method of factoring by grouping. We have shown that the factored form of the expression is $(x - 5)(x + 7)$. This is a crucial skill in algebra, as it helps us simplify complex equations and solve problems more efficiently.

Answer

The correct answer is:

A. $x - 5$

Why is Factoring Important?

Factoring quadratic expressions is an important skill in algebra because it helps us simplify complex equations and solve problems more efficiently. By factoring a quadratic expression, we can:

• Simplify complex equations
• Solve problems more efficiently
• Understand the structure of the equation
• Make it easier to solve the equation

Real-World Applications

Factoring quadratic expressions has many real-world applications, including:

• Physics: Factoring quadratic expressions is used to solve problems involving motion and energy.
• Engineering: Factoring quadratic expressions is used to design and optimize systems.
• Economics: Factoring quadratic expressions is used to model and analyze economic systems.

Tips and Tricks

Here are some tips and tricks to help you factor quadratic expressions:

• Use the method of factoring by grouping to factor quadratic expressions.
• Look for common factors in the terms.
• Use the distributive property to expand the expression.
• Check your work by multiplying the factors together.

Conclusion

Introduction

In our previous article, we discussed the importance of factoring quadratic expressions and provided a step-by-step guide on how to factor the quadratic expression $x^2 + 2x - 35$. In this article, we will answer some frequently asked questions about factoring quadratic expressions.

Q&A

Q: What is the difference between factoring and simplifying a quadratic expression?

A: Factoring a quadratic expression involves expressing it as a product of two binomials, while simplifying a quadratic expression involves reducing it to its simplest form.

Q: How do I know if a quadratic expression can be factored?

A: A quadratic expression can be factored if it can be expressed as a product of two binomials. To determine if a quadratic expression can be factored, look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: What is the method of factoring by grouping?

A: The method of factoring by grouping involves factoring a quadratic expression by grouping the terms into two pairs and then factoring out the common terms.

Q: How do I factor a quadratic expression using the method of factoring by grouping?

A: To factor a quadratic expression using the method of factoring by grouping, follow these steps:

1. Group the terms into two pairs.
2. Factor out the common terms from each pair.
3. Write the factored form of the expression.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that the product of a number and a sum is equal to the sum of the products.

Q: How do I use the distributive property to factor a quadratic expression?

A: To use the distributive property to factor a quadratic expression, follow these steps:

1. Multiply the first term of the first binomial by the second term of the second binomial.
2. Multiply the second term of the first binomial by the second term of the second binomial.
3. Write the factored form of the expression.

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

A: Some common mistakes to avoid when factoring quadratic expressions include:

• Not checking if the expression can be factored.
• Not using the method of factoring by grouping.
• Not using the distributive property.
• Not checking the work by multiplying the factors together.

Q: How do I check my work when factoring a quadratic expression?

A: To check your work when factoring a quadratic expression, follow these steps:

1. Multiply the factors together.
2. Simplify the expression.
3. Check if the original expression is equal to the simplified expression.

Conclusion

In conclusion, factoring quadratic expressions is a crucial skill in algebra that helps us simplify complex equations and solve problems more efficiently. By understanding the method of factoring by grouping and using the distributive property, we can factor quadratic expressions and solve problems more efficiently. We hope this Q&A guide has helped you understand the concept of factoring quadratic expressions and how to apply it in different situations.

Tips and Tricks

Here are some tips and tricks to help you factor quadratic expressions:

• Practice, practice, practice! The more you practice, the better you will become at factoring quadratic expressions.
• Use the method of factoring by grouping to factor quadratic expressions.
• Use the distributive property to factor quadratic expressions.
• Check your work by multiplying the factors together.
• Use a calculator to check your work if you are unsure.

Real-World Applications

Factoring quadratic expressions has many real-world applications, including:

• Physics: Factoring quadratic expressions is used to solve problems involving motion and energy.
• Engineering: Factoring quadratic expressions is used to design and optimize systems.
• Economics: Factoring quadratic expressions is used to model and analyze economic systems.

Conclusion

In conclusion, factoring quadratic expressions is a crucial skill in algebra that helps us simplify complex equations and solve problems more efficiently. By understanding the method of factoring by grouping and using the distributive property, we can factor quadratic expressions and solve problems more efficiently. We hope this Q&A guide has helped you understand the concept of factoring quadratic expressions and how to apply it in different situations.