Factoring $x^2 + 30x + 200$ A Step By Step Guide

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In mathematics, particularly in algebra, factoring quadratic expressions is a fundamental skill. Quadratic expressions, those of the form ax² + bx + c, frequently appear in various mathematical contexts, including solving equations, graphing functions, and simplifying expressions. Mastering the art of factoring not only enhances problem-solving capabilities but also provides a deeper understanding of algebraic structures. In this article, we will dissect the quadratic expression x2+30x+200x^2 + 30x + 200 and explore the process of factoring it, while also addressing the multiple-choice question presented.

Understanding Quadratic Expressions

Before we dive into the specifics of factoring x2+30x+200x^2 + 30x + 200, let's establish a solid understanding of quadratic expressions in general. A quadratic expression is a polynomial expression of degree two, meaning the highest power of the variable is two. The general form of a quadratic expression is ax² + bx + c, where a, b, and c are constants and a ≠ 0. The coefficient a is the quadratic coefficient, b is the linear coefficient, and c is the constant term.

In our case, the quadratic expression x2+30x+200x^2 + 30x + 200 fits this form perfectly. Here, a = 1, b = 30, and c = 200. Understanding these coefficients is crucial for factoring the expression effectively.

The Process of Factoring Quadratic Expressions

Factoring a quadratic expression involves rewriting it as a product of two linear expressions. This process hinges on finding two numbers that satisfy specific conditions related to the coefficients of the quadratic expression. The primary technique for factoring involves identifying two numbers that, when multiplied together, yield the constant term (c) and, when added together, yield the linear coefficient (b).

Let's break down the steps for factoring x2+30x+200x^2 + 30x + 200:

  1. Identify the coefficients: In the given expression, a = 1, b = 30, and c = 200.

  2. Find two numbers: We need to find two numbers that multiply to 200 and add up to 30. This is the core of the factoring process. We can systematically list factor pairs of 200 and check their sums:

    • 1 and 200 (sum = 201)
    • 2 and 100 (sum = 102)
    • 4 and 50 (sum = 54)
    • 5 and 40 (sum = 45)
    • 8 and 25 (sum = 33)
    • 10 and 20 (sum = 30)

    We have found our numbers! The pair 10 and 20 satisfy the conditions.

  3. Rewrite the expression: Using the numbers found, we can rewrite the middle term (30x) as the sum of 10x and 20x:

    x2+30x+200=x2+10x+20x+200x^2 + 30x + 200 = x^2 + 10x + 20x + 200

  4. Factor by grouping: Now, we factor by grouping the first two terms and the last two terms:

    x2+10x+20x+200=x(x+10)+20(x+10)x^2 + 10x + 20x + 200 = x(x + 10) + 20(x + 10)

  5. Final factorization: Notice that both terms have a common factor of (x + 10). We factor this out:

    x(x+10)+20(x+10)=(x+10)(x+20)x(x + 10) + 20(x + 10) = (x + 10)(x + 20)

Thus, the factored form of x2+30x+200x^2 + 30x + 200 is (x + 10)(x + 20).

Analyzing the Multiple-Choice Options

Now that we have factored the quadratic expression, let's evaluate the multiple-choice options provided:

(A) (x+4)(x+50)(x+4)(x+50) (B) (x+5)(x+25)(x+5)(x+25) (C) (x+8)(x+25)(x+8)(x+25) (D) (x+10)(x+20)(x+10)(x+20)

By comparing our factored form (x + 10)(x + 20) with the options, we can clearly see that option (D) is the correct answer.

Expanding and Verifying

To ensure our factorization is correct, we can expand the factored form and verify that it matches the original quadratic expression. Let's expand (x + 10)(x + 20) using the distributive property (also known as the FOIL method):

(x+10)(x+20)=x(x+20)+10(x+20)(x + 10)(x + 20) = x(x + 20) + 10(x + 20)

=x2+20x+10x+200= x^2 + 20x + 10x + 200

=x2+30x+200= x^2 + 30x + 200

As we can see, the expanded form matches the original quadratic expression, confirming that our factorization is correct.

Common Mistakes to Avoid

Factoring quadratic expressions can sometimes be tricky, and certain common mistakes can lead to incorrect results. Here are a few pitfalls to watch out for:

  • Incorrectly identifying factors: Failing to find the correct pair of numbers that multiply to c and add up to b is a common error. Systematic listing and checking of factor pairs can help avoid this.
  • Sign errors: Pay close attention to the signs of the coefficients. Incorrectly applying signs can lead to wrong factorizations. For instance, if c is positive and b is negative, both factors should be negative.
  • Incorrect grouping: When factoring by grouping, ensure that the common factors are correctly identified and factored out. A mistake in this step can derail the entire process.
  • Skipping verification: Always verify your factorization by expanding the factored form and comparing it to the original expression. This simple step can catch errors and ensure accuracy.

Alternative Methods for Factoring

While the method described above is the most common, alternative approaches can be used to factor quadratic expressions, particularly when the coefficients are more complex. These include:

  • The quadratic formula: The quadratic formula can be used to find the roots of a quadratic equation, which can then be used to determine the factors. For a quadratic equation ax² + bx + c = 0, the roots are given by:

    x = rac{-b ext{±} ext{√}(b^2 - 4ac)}{2a}

    If the roots are r₁ and r₂, then the quadratic expression can be factored as a(x - r₁)(x - r₂).

  • Completing the square: This method involves manipulating the quadratic expression to create a perfect square trinomial, which can then be factored easily.

Real-World Applications of Factoring

Factoring quadratic expressions is not merely an academic exercise; it has numerous real-world applications across various fields, including:

  • Physics: Quadratic equations and factoring are used to solve problems related to projectile motion, energy, and mechanics.
  • Engineering: Engineers use quadratic equations in structural design, electrical circuits, and control systems.
  • Economics: Quadratic functions are used to model cost, revenue, and profit in business and economics.
  • Computer graphics: Quadratic equations play a role in creating curves and surfaces in computer graphics and animation.
  • Optimization problems: Many optimization problems in mathematics and computer science involve finding the minimum or maximum of a quadratic function, which often requires factoring.

Conclusion

Factoring the quadratic expression x2+30x+200x^2 + 30x + 200 involves identifying two numbers that multiply to the constant term (200) and add up to the linear coefficient (30). By systematically finding these numbers (10 and 20), we successfully factored the expression as (x + 10)(x + 20). This factorization aligns with option (D) in the multiple-choice question.

Mastering the techniques of factoring quadratic expressions is crucial for success in algebra and beyond. By understanding the underlying principles and practicing consistently, you can confidently tackle a wide range of factoring problems and appreciate the versatility of this fundamental algebraic skill. Remember to always verify your factorizations and be mindful of common mistakes. Factoring is a cornerstone of mathematical problem-solving, and its applications extend far beyond the classroom.

This comprehensive guide has not only provided a step-by-step solution to the given problem but also offered a broader understanding of factoring quadratic expressions, common pitfalls, alternative methods, and real-world applications. By delving into these aspects, we aim to equip you with the knowledge and skills to excel in your mathematical journey.