What Is The Factored Form Of $x^2 - X - 2$?A. $(x-2)(x+1)$B. $ ( X + 2 ) ( X + 1 ) (x+2)(x+1) ( X + 2 ) ( X + 1 ) [/tex]C. $(x-2)(x-1)$D. $(x+2)(x-1)$

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Introduction to Factoring Quadratic Expressions

Factoring quadratic expressions is a crucial concept in algebra, and it involves expressing a quadratic expression as a product of two binomial expressions. This process is essential in solving quadratic equations, simplifying expressions, and understanding the properties of quadratic functions. In this article, we will focus on finding the factored form of the quadratic expression $x^2 - x - 2$.

Understanding the Concept of Factoring

Factoring a quadratic expression involves finding two binomial expressions whose product equals the original quadratic expression. The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. To factor a quadratic expression, we need to find two numbers whose product is $ac$ and whose sum is $b$. These numbers are called the factors of the quadratic expression.

Factoring the Quadratic Expression $x^2 - x - 2$

To factor the quadratic expression $x^2 - x - 2$, we need to find two numbers whose product is $-2$ and whose sum is $-1$. These numbers are $-2$ and $1$, since $(-2) \times (1) = -2$ and $(-2) + (1) = -1$. Therefore, we can write the quadratic expression as $(x-2)(x+1)$.

Verifying the Factored Form

To verify that the factored form $(x-2)(x+1)$ is correct, we can multiply the two binomial expressions together. This will give us the original quadratic expression $x^2 - x - 2$. Multiplying $(x-2)(x+1)$, we get:

(x−2)(x+1)=x2+x−2x−2=x2−x−2(x-2)(x+1) = x^2 + x - 2x - 2 = x^2 - x - 2

This confirms that the factored form $(x-2)(x+1)$ is correct.

Conclusion

In conclusion, the factored form of the quadratic expression $x^2 - x - 2$ is $(x-2)(x+1)$. This involves finding two numbers whose product is $-2$ and whose sum is $-1$, and then writing the quadratic expression as a product of two binomial expressions. The factored form is essential in solving quadratic equations, simplifying expressions, and understanding the properties of quadratic functions.

Comparison with Other Options

Let's compare the factored form $(x-2)(x+1)$ with the other options:

  • Option A: $(x+2)(x+1)$
  • Option B: $(x+2)(x-1)$
  • Option C: $(x-2)(x-1)$
  • Option D: $(x+2)(x-1)$

We can see that only option A, $(x+2)(x+1)$, is not equal to the factored form $(x-2)(x+1)$. The other options are not correct.

Final Answer

The final answer is:

A. $(x-2)(x+1)$

This is the correct factored form of the quadratic expression $x^2 - x - 2$.

Introduction

Factoring quadratic expressions is a crucial concept in algebra, and it involves expressing a quadratic expression as a product of two binomial expressions. In our previous article, we discussed how to factor the quadratic expression $x^2 - x - 2$. In this article, we will provide a Q&A guide to help you understand the concept of factoring quadratic expressions and how to apply it to different types of quadratic expressions.

Q: What is the general form of a quadratic expression?

A: The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What is factoring a quadratic expression?

A: Factoring a quadratic expression involves finding two binomial expressions whose product equals the original quadratic expression.

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

A: To find the factored form of a quadratic expression, you need to find two numbers whose product is $ac$ and whose sum is $b$. These numbers are called the factors of the quadratic expression.

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

A: The steps to factor a quadratic expression are:

  1. Identify the values of $a$, $b$, and $c$ in the quadratic expression.
  2. Find two numbers whose product is $ac$ and whose sum is $b$.
  3. Write the quadratic expression as a product of two binomial expressions using the numbers found in step 2.

Q: How do I verify the factored form of a quadratic expression?

A: To verify the factored form of a quadratic expression, you can multiply the two binomial expressions together. This will give you the original quadratic 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 identifying the values of $a$, $b$, and $c$ correctly.
  • Not finding the correct numbers whose product is $ac$ and whose sum is $b$.
  • Not writing the quadratic expression as a product of two binomial expressions correctly.

Q: Can you provide examples of factoring quadratic expressions?

A: Yes, here are some examples of factoring quadratic expressions:

  • Factoring $x^2 + 5x + 6$:
    • Identify the values of $a$, $b$, and $c$: $a = 1$, $b = 5$, $c = 6$.
    • Find two numbers whose product is $ac$ and whose sum is $b$: $2$ and $3$.
    • Write the quadratic expression as a product of two binomial expressions: $(x+2)(x+3)$.
  • Factoring $x^2 - 7x - 18$:
    • Identify the values of $a$, $b$, and $c$: $a = 1$, $b = -7$, $c = -18$.
    • Find two numbers whose product is $ac$ and whose sum is $b$: $-9$ and $2$.
    • Write the quadratic expression as a product of two binomial expressions: $(x-9)(x+2)$.

Q: Can you provide a summary of the key concepts in factoring quadratic expressions?

A: Yes, here is a summary of the key concepts in factoring quadratic expressions:

  • The general form of a quadratic expression is $ax^2 + bx + c$.
  • Factoring a quadratic expression involves finding two binomial expressions whose product equals the original quadratic expression.
  • To find the factored form of a quadratic expression, you need to find two numbers whose product is $ac$ and whose sum is $b$.
  • The steps to factor a quadratic expression are: identify the values of $a$, $b$, and $c$, find two numbers whose product is $ac$ and whose sum is $b$, and write the quadratic expression as a product of two binomial expressions using the numbers found.

Conclusion

In conclusion, factoring quadratic expressions is a crucial concept in algebra, and it involves expressing a quadratic expression as a product of two binomial expressions. By understanding the key concepts and following the steps outlined in this article, you can master the art of factoring quadratic expressions and apply it to different types of quadratic expressions.