Factor: $x^2 - 2x - 15$A. $(x - 5)(x + 3)$ B. \$(x - 2)(x + 5)$[/tex\] C. $(x + 5)(x - 3)$ D. $(x - 5)(x + 2)$

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Introduction

Factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. This technique is essential in solving quadratic equations, simplifying expressions, and understanding the properties of quadratic functions. In this article, we will explore the process of factoring quadratic expressions, with a focus on the given quadratic expression: $x^2 - 2x - 15$.

Understanding Quadratic Expressions

A quadratic expression is a polynomial of degree two, which means it has the highest power of the variable (in this case, x) equal to two. The general form of a quadratic expression is $ax^2 + bx + c$, where a, b, and c are constants. In the given expression, $x^2 - 2x - 15$, we have a = 1, b = -2, and c = -15.

Factoring Quadratic Expressions

Factoring a quadratic expression involves expressing it as a product of two binomials. The process of factoring involves finding two numbers whose product is equal to the constant term (c) and whose sum is equal to the coefficient of the linear term (b). In this case, we need to find two numbers whose product is -15 and whose sum is -2.

Finding the Factors

To find the factors, we can start by listing the factors of -15:

  • 1 and -15
  • -1 and 15
  • 3 and -5
  • -3 and 5

We can then check which pair of factors adds up to -2:

  • 1 + (-15) = -14 (not equal to -2)
  • -1 + 15 = 14 (not equal to -2)
  • 3 + (-5) = -2 (matches the given condition)
  • -3 + 5 = 2 (not equal to -2)

Factoring the Quadratic Expression

Based on the factors we found, we can write the quadratic expression as a product of two binomials:

x2−2x−15=(x−5)(x+3)x^2 - 2x - 15 = (x - 5)(x + 3)

This is the factored form of the given quadratic expression.

Checking the Answer

To verify that the factored form is correct, we can multiply the two binomials:

(x−5)(x+3)=x2+3x−5x−15(x - 5)(x + 3) = x^2 + 3x - 5x - 15

Simplifying the expression, we get:

x2−2x−15x^2 - 2x - 15

This matches the original quadratic expression, confirming that the factored form is correct.

Conclusion

In this article, we explored the process of factoring quadratic expressions, with a focus on the given quadratic expression: $x^2 - 2x - 15$. We found the factors of the expression and wrote it as a product of two binomials. The factored form is $(x - 5)(x + 3)$, which can be verified by multiplying the two binomials.

Common Mistakes to Avoid

When factoring quadratic expressions, it's essential to avoid common mistakes. Here are a few:

  • Not checking the factors: Make sure to check the factors of the constant term and the sum of the factors.
  • Not writing the factored form correctly: Double-check that the factored form is written correctly, with the correct signs and coefficients.
  • Not verifying the answer: Verify the answer by multiplying the two binomials to ensure that it matches the original quadratic expression.

Practice Problems

To practice factoring quadratic expressions, try the following problems:

  • Factor the quadratic expression: $x^2 + 5x + 6$
  • Factor the quadratic expression: $x^2 - 7x - 18$
  • Factor the quadratic expression: $x^2 + 2x - 15$

Real-World Applications

Factoring quadratic expressions has numerous real-world applications. Here are a few:

  • Solving quadratic equations: Factoring quadratic expressions is essential in solving quadratic equations, which are used in various fields, such as physics, engineering, and economics.
  • Simplifying expressions: Factoring quadratic expressions can help simplify complex expressions, making them easier to work with.
  • Understanding quadratic functions: Factoring quadratic expressions can help understand the properties of quadratic functions, which are used in various fields, such as physics, engineering, and economics.

Conclusion

In conclusion, factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. The process of factoring involves finding two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term. By following the steps outlined in this article, you can factor quadratic expressions with ease.

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Introduction

Factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. In our previous article, we explored the process of factoring quadratic expressions, with a focus on the given quadratic expression: $x^2 - 2x - 15$. 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 combining like terms to reduce the expression 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 equal to the constant term and whose sum is equal to the coefficient of the linear term.

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 the factors of the constant term and the sum of the factors
  • Not writing the factored form correctly, with the correct signs and coefficients
  • Not verifying the answer by multiplying the two binomials to ensure that it matches the original quadratic expression

Q: How do I factor a quadratic expression with a negative leading coefficient?

A: To factor a quadratic expression with a negative leading coefficient, follow the same steps as factoring a quadratic expression with a positive leading coefficient. The only difference is that the factored form will have a negative sign in front of one of the binomials.

Q: Can a quadratic expression have more than two factors?

A: Yes, a quadratic expression can have more than two factors. However, in most cases, a quadratic expression can be factored into two binomials.

Q: How do I factor a quadratic expression with a coefficient of 1 in front of the x^2 term?

A: To factor a quadratic expression with a coefficient of 1 in front of the x^2 term, follow the same steps as factoring a quadratic expression with a coefficient of 1 in front of the x term. The only difference is that the factored form will have a coefficient of 1 in front of the x term.

Q: Can a quadratic expression be factored if it has a complex coefficient?

A: Yes, a quadratic expression can be factored if it has a complex coefficient. However, the factored form will involve complex numbers.

Q: How do I factor a quadratic expression with a coefficient of 0 in front of the x term?

A: To factor a quadratic expression with a coefficient of 0 in front of the x term, follow the same steps as factoring a quadratic expression with a coefficient of 1 in front of the x term. The only difference is that the factored form will have a coefficient of 0 in front of the x term.

Conclusion

In conclusion, factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. By following the steps outlined in this article, you can answer frequently asked questions about factoring quadratic expressions.

Practice Problems

To practice factoring quadratic expressions, try the following problems:

  • Factor the quadratic expression: $x^2 + 5x + 6$
  • Factor the quadratic expression: $x^2 - 7x - 18$
  • Factor the quadratic expression: $x^2 + 2x - 15$

Real-World Applications

Factoring quadratic expressions has numerous real-world applications. Here are a few:

  • Solving quadratic equations: Factoring quadratic expressions is essential in solving quadratic equations, which are used in various fields, such as physics, engineering, and economics.
  • Simplifying expressions: Factoring quadratic expressions can help simplify complex expressions, making them easier to work with.
  • Understanding quadratic functions: Factoring quadratic expressions can help understand the properties of quadratic functions, which are used in various fields, such as physics, engineering, and economics.

Conclusion

In conclusion, factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. By following the steps outlined in this article, you can factor quadratic expressions with ease.