Factoring Quadratic Expressions X² + 15x + 56 A Step By Step Guide

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Factoring quadratic expressions is a fundamental skill in algebra, and mastering it opens doors to solving various mathematical problems. In this comprehensive guide, we'll delve into the process of factoring the quadratic expression x² + 15x + 56, providing a clear and concise approach that will empower you to tackle similar problems with confidence. We'll break down the steps involved, explain the underlying concepts, and provide illustrative examples to solidify your understanding. So, let's embark on this journey of algebraic exploration and unlock the secrets of factoring quadratic expressions!

Understanding Quadratic Expressions

Before we dive into the factoring process, it's essential to grasp the concept of quadratic expressions. A quadratic expression is a polynomial expression of the form ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'x' is a variable. The term 'quadratic' stems from the fact that the highest power of the variable 'x' is 2. In our specific case, the expression x² + 15x + 56 fits this form, with a = 1, b = 15, and c = 56.

Key characteristics of quadratic expressions:

  • They involve a variable raised to the power of 2 (x²). This is the hallmark of a quadratic expression.
  • They can also include terms with the variable raised to the power of 1 (x) and a constant term (a number without a variable).
  • The coefficients (the numbers multiplying the variables) and the constant term can be positive, negative, or zero.

Why is factoring important?

Factoring a quadratic expression means rewriting it as a product of two linear expressions (expressions of the form px + q, where p and q are constants). This process is crucial for several reasons:

  • Solving quadratic equations: Factoring allows us to find the values of 'x' that make the expression equal to zero. These values are known as the roots or solutions of the quadratic equation.
  • Simplifying algebraic expressions: Factoring can help us simplify complex expressions and make them easier to work with.
  • Graphing quadratic functions: The factored form of a quadratic expression provides valuable information about the graph of the corresponding quadratic function, such as its x-intercepts.

The Factoring Process: A Step-by-Step Approach

Now that we have a solid understanding of quadratic expressions and the importance of factoring, let's delve into the step-by-step process of factoring x² + 15x + 56.

Step 1: Identify the coefficients and the constant term.

In our expression, x² + 15x + 56, we have:

  • Coefficient of x² (a) = 1
  • Coefficient of x (b) = 15
  • Constant term (c) = 56

Step 2: Find two numbers that multiply to 'c' and add up to 'b'.

This is the heart of the factoring process. We need to find two numbers, let's call them 'p' and 'q', such that:

  • p * q = c (p multiplied by q equals the constant term)
  • p + q = b (p plus q equals the coefficient of x)

In our case, we need to find two numbers that multiply to 56 and add up to 15. Let's systematically list the pairs of factors of 56:

  • 1 and 56 (1 * 56 = 56, 1 + 56 = 57)
  • 2 and 28 (2 * 28 = 56, 2 + 28 = 30)
  • 4 and 14 (4 * 14 = 56, 4 + 14 = 18)
  • 7 and 8 (7 * 8 = 56, 7 + 8 = 15)

We've found our numbers! 7 and 8 satisfy both conditions: they multiply to 56 and add up to 15.

Step 3: Rewrite the expression using the numbers found in Step 2.

Now we rewrite the original expression, x² + 15x + 56, by splitting the middle term (15x) using the numbers we found (7 and 8):

x² + 15x + 56 = x² + 7x + 8x + 56

Step 4: Factor by grouping.

We now have four terms. We'll group the first two terms and the last two terms and factor out the greatest common factor (GCF) from each group:

  • From the first group (x² + 7x), the GCF is 'x'. Factoring out 'x', we get: x(x + 7)
  • From the second group (8x + 56), the GCF is '8'. Factoring out '8', we get: 8(x + 7)

Now our expression looks like this:

x(x + 7) + 8(x + 7)

Notice that both terms have a common factor of (x + 7). We can factor this out:

(x + 7)(x + 8)

Step 5: Write the factored form.

We have successfully factored the quadratic expression! The factored form of x² + 15x + 56 is (x + 7)(x + 8).

The Answer: (x + 7)(x + 8)

Therefore, the factored form of the expression x² + 15x + 56 is (x + 7)(x + 8). We have successfully applied the steps of factoring to arrive at the correct answer. Remember, the key is to find the two numbers that multiply to the constant term and add up to the coefficient of the x term. Practice is crucial to master this skill, so let's explore some more examples to solidify your understanding.

Substituting Numerical Values: Verifying the Factored Form

To ensure the factored form is correct, we can substitute numerical values for 'x' in both the original expression and the factored form and verify that they yield the same result. This process is a valuable check for accuracy and helps build confidence in your factoring skills.

Let's substitute x = 2 into the original expression:

x² + 15x + 56 = (2)² + 15(2) + 56 = 4 + 30 + 56 = 90

Now, let's substitute x = 2 into the factored form:

(x + 7)(x + 8) = (2 + 7)(2 + 8) = (9)(10) = 90

As we can see, both the original expression and the factored form evaluate to 90 when x = 2. This provides strong evidence that our factoring is correct. We can repeat this process with different values of 'x' to further confirm our result.

Let's try x = -3:

  • Original expression: (-3)² + 15(-3) + 56 = 9 - 45 + 56 = 20
  • Factored form: (-3 + 7)(-3 + 8) = (4)(5) = 20

Again, both expressions yield the same result, reinforcing the correctness of our factored form.

Common Mistakes to Avoid

Factoring quadratic expressions can be tricky, and it's essential to be aware of common mistakes to avoid them. Here are some pitfalls to watch out for:

  • Incorrectly identifying the signs: Pay close attention to the signs of the coefficients and the constant term. A mistake in the signs can lead to incorrect factoring.
  • Not finding the correct pair of numbers: The key to factoring is finding the two numbers that multiply to 'c' and add up to 'b'. Make sure you systematically consider all possible pairs of factors.
  • Factoring out the GCF incorrectly: When factoring by grouping, ensure you factor out the greatest common factor (GCF) correctly from each group.
  • Stopping too early: Remember, the goal is to rewrite the expression as a product of two linear expressions. Don't stop until you have achieved this.

By being mindful of these common mistakes, you can significantly improve your accuracy and confidence in factoring quadratic expressions.

Practice Problems

To solidify your understanding and enhance your factoring skills, let's tackle a few more practice problems:

  1. Factor x² + 10x + 24
  2. Factor x² - 5x - 14
  3. Factor x² + 2x - 15

Work through these problems using the step-by-step approach we discussed earlier. Check your answers by substituting numerical values or by expanding the factored form to see if it matches the original expression.

Conclusion

Factoring quadratic expressions is a fundamental skill in algebra, and mastering it is crucial for solving equations, simplifying expressions, and understanding the behavior of quadratic functions. In this guide, we've provided a comprehensive step-by-step approach to factoring the expression x² + 15x + 56. We've also discussed the importance of factoring, common mistakes to avoid, and practice problems to further enhance your skills. With consistent practice and a clear understanding of the concepts, you'll be well-equipped to tackle any quadratic expression that comes your way. So, keep practicing, keep exploring, and keep unlocking the power of algebra!

By understanding factoring quadratic expressions like x² + 15x + 56, you're building a strong foundation in algebra, which is essential for more advanced mathematical concepts. The ability to break down complex expressions into simpler forms is a valuable skill that will serve you well in various mathematical contexts. Remember, practice is key to mastery, so keep working on factoring problems, and you'll see your skills grow over time. This method of finding two numbers that multiply to the constant term and add up to the coefficient of the linear term is a fundamental technique in algebra, and it forms the basis for solving many types of quadratic equations and problems. Mastering factoring techniques can significantly boost your confidence and proficiency in algebra. Factoring quadratic expressions such as this one is a cornerstone of algebraic problem-solving.