Complete The Factorization Of The Quadratic Expression: $\[ X^2 + 2x - 24 = (x + [?])(x - [?]) \\]

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Introduction

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In algebra, factorization is a process of expressing a polynomial as a product of simpler polynomials. Quadratic expressions are a type of polynomial that can be factored into the product of two binomials. In this article, we will learn how to complete the factorization of the quadratic expression $x^2 + 2x - 24$.

What is Factorization?

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Factorization is a process of expressing a polynomial as a product of simpler polynomials. It involves finding the factors of the polynomial, which are the numbers or expressions that multiply together to give the original polynomial. Factorization is an important concept in algebra, as it allows us to simplify complex expressions and solve equations.

Types of Factorization

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There are several types of factorization, including:

• Linear factorization: This involves expressing a polynomial as the product of two linear factors.
• Quadratic factorization: This involves expressing a polynomial as the product of two quadratic factors.
• Cubic factorization: This involves expressing a polynomial as the product of three linear factors.

How to Factorize a Quadratic Expression

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To factorize a quadratic expression, we need to find two numbers whose product is equal to the constant term of the expression and whose sum is equal to the coefficient of the linear term. These numbers are called the factors of the quadratic expression.

Step 1: Identify the Coefficients

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The first step in factorizing a quadratic expression is to identify the coefficients of the terms. In the expression $x^2 + 2x - 24$, the coefficients are:

• Coefficient of $x^2$: 1
• Coefficient of $x$: 2
• Constant term: -24

Step 2: Find the Factors

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The next step is to find the factors of the quadratic expression. We need to find two numbers whose product is equal to the constant term (-24) and whose sum is equal to the coefficient of the linear term (2).

Step 3: Write the Factors as Binomials

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Once we have found the factors, we can write them as binomials. In this case, the factors are 6 and -4, so we can write the quadratic expression as:

$$x^2 + 2x - 24 = (x + 6)(x - 4)$$

Example

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Let's consider an example to illustrate the process of factorizing a quadratic expression. Suppose we want to factorize the expression $x^2 + 5x - 6$. We can follow the steps outlined above to find the factors and write the expression as a product of binomials.

Step 1: Identify the Coefficients

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The coefficients of the expression $x^2 + 5x - 6$ are:

• Coefficient of $x^2$: 1
• Coefficient of $x$: 5
• Constant term: -6

Step 2: Find the Factors

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We need to find two numbers whose product is equal to the constant term (-6) and whose sum is equal to the coefficient of the linear term (5). The factors of -6 are:

• Factors: 6 and -1

Step 3: Write the Factors as Binomials

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We can write the factors as binomials as follows:

$$x^2 + 5x - 6 = (x + 6)(x - 1)$$

Conclusion

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In this article, we have learned how to complete the factorization of the quadratic expression $x^2 + 2x - 24$. We have also seen how to factorize a quadratic expression using the steps outlined above. Factorization is an important concept in algebra, as it allows us to simplify complex expressions and solve equations.

Frequently Asked Questions

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Q: What is factorization?

A: Factorization is a process of expressing a polynomial as a product of simpler polynomials.

Q: How do I factorize a quadratic expression?

A: To factorize a quadratic expression, you need to find two numbers whose product is equal to the constant term of the expression and whose sum is equal to the coefficient of the linear term.

Q: What are the types of factorization?

A: There are several types of factorization, including linear factorization, quadratic factorization, and cubic factorization.

Q: How do I write the factors as binomials?

A: Once you have found the factors, you can write them as binomials by multiplying the factors together.

Further Reading

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If you want to learn more about factorization, I recommend checking out the following resources:

• Algebra textbooks: There are many algebra textbooks that cover factorization in detail.
• Online resources: There are many online resources that provide tutorials and examples on factorization.
• Math websites: There are many math websites that provide information and resources on factorization.

References

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• Algebra textbooks: There are many algebra textbooks that cover factorization in detail.
• Online resources: There are many online resources that provide tutorials and examples on factorization.
• Math websites: There are many math websites that provide information and resources on factorization.
  

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Introduction

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In our previous article, we learned how to complete the factorization of the quadratic expression $x^2 + 2x - 24$. In this article, we will answer some frequently asked questions about quadratic factorization.

Q&A

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Q: What is the difference between factorization and simplification?

A: Factorization is the process of expressing a polynomial as a product of simpler polynomials, while simplification is the process of combining like terms to simplify an expression.

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 the product of two binomials. To determine if a quadratic expression can be factored, try to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Q: What are the steps to factorize a quadratic expression?

A: The steps to factorize a quadratic expression are:

1. Identify the coefficients of the terms.
2. Find the factors of the quadratic expression.
3. Write the factors as binomials.

Q: How do I find the factors of a quadratic expression?

A: To find the factors of a quadratic expression, you need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Q: What are the types of factorization?

A: There are several types of factorization, including:

• Linear factorization: This involves expressing a polynomial as the product of two linear factors.
• Quadratic factorization: This involves expressing a polynomial as the product of two quadratic factors.
• Cubic factorization: This involves expressing a polynomial as the product of three linear factors.

Q: How do I write the factors as binomials?

A: Once you have found the factors, you can write them as binomials by multiplying the factors together.

Q: What are some common mistakes to avoid when factorizing a quadratic expression?

A: Some common mistakes to avoid when factorizing a quadratic expression include:

• Not identifying the coefficients correctly: Make sure to identify the coefficients of the terms correctly.
• Not finding the correct factors: Make sure to find the correct factors of the quadratic expression.
• Not writing the factors as binomials correctly: Make sure to write the factors as binomials correctly.

Example

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Let's consider an example to illustrate the process of factorizing a quadratic expression. Suppose we want to factorize the expression $x^2 + 7x + 12$. We can follow the steps outlined above to find the factors and write the expression as a product of binomials.

Step 1: Identify the Coefficients

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The coefficients of the expression $x^2 + 7x + 12$ are:

• Coefficient of $x^2$: 1
• Coefficient of $x$: 7
• Constant term: 12

Step 2: Find the Factors

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We need to find two numbers whose product is equal to the constant term (12) and whose sum is equal to the coefficient of the linear term (7). The factors of 12 are:

• Factors: 3 and 4

Step 3: Write the Factors as Binomials

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We can write the factors as binomials as follows:

$$x^2 + 7x + 12 = (x + 3)(x + 4)$$

Conclusion

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In this article, we have answered some frequently asked questions about quadratic factorization. We have also seen how to factorize a quadratic expression using the steps outlined above. Factorization is an important concept in algebra, as it allows us to simplify complex expressions and solve equations.

Further Reading

---

If you want to learn more about factorization, I recommend checking out the following resources:

• Algebra textbooks: There are many algebra textbooks that cover factorization in detail.
• Online resources: There are many online resources that provide tutorials and examples on factorization.
• Math websites: There are many math websites that provide information and resources on factorization.

References

---

• Algebra textbooks: There are many algebra textbooks that cover factorization in detail.
• Online resources: There are many online resources that provide tutorials and examples on factorization.
• Math websites: There are many math websites that provide information and resources on factorization.