Factor 2x 2 13x 15

Factoring the Quadratic Expression 2x² + 13x + 15: A Comprehensive Guide

Factoring quadratic expressions is a fundamental skill in algebra. Understanding how to factor allows you to solve quadratic equations, simplify complex expressions, and delve deeper into various mathematical concepts. This article provides a thorough explanation of how to factor the quadratic expression 2x² + 13x + 15, covering multiple methods and offering insights into the underlying mathematical principles. We'll explore different approaches, ensuring a clear understanding for learners of all levels.

Understanding Quadratic Expressions

Before diving into the factoring process, let's refresh our understanding of quadratic expressions. A quadratic expression is a polynomial of degree two, meaning the highest power of the variable (usually 'x') is 2. It generally takes the form ax² + bx + c, where 'a', 'b', and 'c' are constants. In our case, the quadratic expression is 2x² + 13x + 15, where a = 2, b = 13, and c = 15. Our goal is to rewrite this expression as a product of two simpler expressions (binomials).

Method 1: AC Method (Factoring by Grouping)

This method is particularly useful when the coefficient of x² (the 'a' value) is not equal to 1. Here's how it works for 2x² + 13x + 15:

1. Find the product 'ac': Multiply the coefficient of x² (a = 2) by the constant term (c = 15). This gives us ac = 2 * 15 = 30.

2. Find two numbers that add up to 'b' and multiply to 'ac': We need two numbers that add up to 13 (the coefficient of x, which is 'b') and multiply to 30. These numbers are 3 and 10 (3 + 10 = 13 and 3 * 10 = 30).

3. Rewrite the middle term: Replace the middle term (13x) with the two numbers we found, ensuring we maintain the original expression's value: 2x² + 3x + 10x + 15.

4. Factor by grouping: Group the terms in pairs and factor out the greatest common factor (GCF) from each pair:

   x(2x + 3) + 5(2x + 3)

5. Factor out the common binomial: Notice that (2x + 3) is common to both terms. Factor it out:

   (2x + 3)(x + 5)

Therefore, the factored form of 2x² + 13x + 15 is (2x + 3)(x + 5).

Method 2: Trial and Error

This method involves directly attempting different combinations of binomial factors until you find the correct one. It's a more intuitive approach but might require more trial and error, especially with larger coefficients.

Since the coefficient of x² is 2, the factors must be of the form (2x + m)(x + n), where 'm' and 'n' are constants. We need to find 'm' and 'n' such that:

• The product of the constant terms (m * n) equals 15.
• The sum of the products of the outer and inner terms (2n + m) equals 13.

Let's try different combinations:

• If m = 1 and n = 15, then 2n + m = 31 (incorrect)
• If m = 3 and n = 5, then 2n + m = 13 (correct!)
• If m = 5 and n = 3, then 2n + m = 11 (incorrect)
• If m = 15 and n = 1, then 2n + m = 17 (incorrect)

The correct combination is m = 3 and n = 5, leading to the factored form (2x + 3)(x + 5).

Method 3: Using the Quadratic Formula (Indirect Factoring)

While not a direct factoring method, the quadratic formula can help find the roots of the quadratic equation 2x² + 13x + 15 = 0. These roots can then be used to determine the factors.

The quadratic formula is:

x = [-b ± √(b² - 4ac)] / 2a

Plugging in the values (a = 2, b = 13, c = 15), we get:

x = [-13 ± √(13² - 4 * 2 * 15)] / (2 * 2) x = [-13 ± √(169 - 120)] / 4 x = [-13 ± √49] / 4 x = (-13 ± 7) / 4

This gives us two solutions:

x₁ = (-13 + 7) / 4 = -6 / 4 = -3/2 x₂ = (-13 - 7) / 4 = -20 / 4 = -5

The factors are then (x - x₁) and (x - x₂):

(x + 3/2) and (x + 5)

To get rid of the fraction, we multiply the first factor by 2:

2(x + 3/2) = (2x + 3)

Thus, the factored form is (2x + 3)(x + 5).

A Deeper Dive: The Significance of Factoring

Factoring quadratic expressions isn't just about manipulating symbols; it reveals crucial information about the underlying quadratic function. The factors represent the x-intercepts of the parabola representing the function y = 2x² + 13x + 15. These intercepts are the points where the graph crosses the x-axis (where y = 0). In this case, the x-intercepts are x = -3/2 and x = -5.

Furthermore, factoring simplifies the expression, making it easier to analyze its behavior, solve equations involving it, and apply it in various mathematical contexts like calculus and differential equations. For instance, solving the equation 2x² + 13x + 15 = 0 becomes significantly easier once factored, as it reduces to solving two simpler linear equations: (2x + 3) = 0 and (x + 5) = 0.

Frequently Asked Questions (FAQ)

• Q: Can I use any method to factor a quadratic expression? A: While all methods discussed will eventually lead to the correct factored form, some are more efficient than others depending on the coefficients of the expression. The AC method is generally preferred when 'a' is not 1. Trial and error is quick for simpler expressions, and the quadratic formula is valuable when factoring proves difficult or impossible.

• Q: What if the quadratic expression is not factorable? A: Not all quadratic expressions can be factored using integers. In such cases, the quadratic formula is still applicable, and the roots will be irrational or complex numbers. The expression can be expressed in its original form or using the completed square form.

• Q: Is there only one correct factored form? A: While the order of the factors might differ, the fundamental factorization remains the same. For example, (2x + 3)(x + 5) and (x + 5)(2x + 3) represent the same factorization.

• Q: How can I check if my factorization is correct? A: The easiest way to check is by expanding the factored form. If the expansion yields the original quadratic expression, then the factorization is correct. In this case, expanding (2x + 3)(x + 5) yields 2x² + 10x + 3x + 15 = 2x² + 13x + 15, confirming our factorization.

Conclusion

Factoring the quadratic expression 2x² + 13x + 15, whether through the AC method, trial and error, or indirectly using the quadratic formula, results in the factored form (2x + 3)(x + 5). This process is not merely an algebraic manipulation but provides key insights into the nature of the quadratic function, simplifying equation solving and furthering our understanding of its behavior. Mastering these techniques is crucial for success in algebra and beyond, paving the way for more advanced mathematical concepts. Remember to practice regularly and explore different methods to solidify your understanding. The more you practice, the more intuitive and efficient your factoring skills will become.