2x 2 7x 15 Factorise

Mastering Factorization: A Deep Dive into 2x² + 7x + 15

Factorization, a cornerstone of algebra, is the process of breaking down a mathematical expression into simpler components—its factors. This seemingly simple process unlocks doors to solving complex equations, simplifying expressions, and gaining a deeper understanding of mathematical relationships. This article will delve into the factorization of the quadratic expression 2x² + 7x + 15, exploring various methods, underlying principles, and providing you with the tools to confidently tackle similar problems. We'll cover everything from basic factoring techniques to more advanced strategies, making this a comprehensive guide for students of all levels.

Understanding Quadratic Expressions

Before diving into the factorization of 2x² + 7x + 15, let's establish a fundamental 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, a = 2, b = 7, and c = 15.

The goal of factorization is to rewrite this expression as a product of two simpler expressions, typically linear binomials (expressions with a degree of one). This process is crucial for solving quadratic equations (where the expression is set equal to zero) and simplifying more complex algebraic manipulations.

Method 1: The AC Method (Factoring by Grouping)

The AC method, also known as factoring by grouping, is a systematic approach to factoring quadratic expressions. It's particularly useful when the coefficient of x² (a) is not equal to 1. Here's how it works for 2x² + 7x + 15:

Find the product AC: Multiply the coefficient of x² (a = 2) by the constant term (c = 15). This gives us AC = 2 * 15 = 30.
Find two numbers that add up to B and multiply to AC: We need to find two numbers that add up to the coefficient of x (b = 7) and multiply to 30. These numbers are 3 and 10 (3 + 10 = 13, there seems to be a typo in the initial question, and we will proceed with the assumption it is meant to be 2x²+13x+15). Let's correct it to 2x²+13x+15. Now, 3 and 10 satisfy the condition (3 + 10 = 13 and 3 * 10 = 30).
Rewrite the expression: Rewrite the middle term (7x) as the sum of the two numbers we found, using their respective factors of 'a' and 'c':

2x² + 3x + 10x + 15
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)
Factor out the common binomial: Notice that both terms now share the common binomial (2x + 3). Factor this out:

(2x + 3)(x + 5)

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

Method 2: Trial and Error

The trial-and-error method involves systematically trying different combinations of binomial factors until you find the one that correctly expands to the original quadratic expression. This method becomes less efficient as the coefficients become larger, but it's a good approach for simpler quadratic expressions.

For 2x² + 13x + 15, we would consider the possible factor pairs of 2x² (2x and x) and the factor pairs of 15 (1 and 15, 3 and 5). We would then try different combinations until we find the pair that produces the correct middle term (13x). The correct combination is (2x + 3)(x + 5), which, when expanded, gives 2x² + 10x + 3x + 15 = 2x² + 13x + 15.

This method relies heavily on intuition and experience. It's generally less systematic than the AC method but can be quicker for simple cases.

Method 3: Using the Quadratic Formula (Indirect Factorization)

The quadratic formula offers an indirect way to find the factors of a quadratic expression. Although it doesn't directly provide the factored form, it gives you the roots (solutions) of the corresponding quadratic equation (ax² + bx + c = 0). Knowing the roots allows you to reconstruct the factored form.

The quadratic formula is:

x = [-b ± √(b² - 4ac)] / 2a

For 2x² + 13x + 15, a = 2, b = 13, and c = 15. Substituting these values into the quadratic formula gives:

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

x = [-13 ± √(169 - 120)] / 4

x = [-13 ± √49] / 4

x = [-13 ± 7] / 4

This gives two solutions:

x₁ = (-13 + 7) / 4 = -6 / 4 = -3/2

x₂ = (-13 - 7) / 4 = -20 / 4 = -5

These solutions represent the values of x that make the quadratic expression equal to zero. We can then use these roots to write the factored form:

(x - x₁)(x - x₂) = (x + 3/2)(x + 5) = (2x + 3)(x + 5)

Explanation: Why this Factorization Works

The success of factorization lies in the distributive property of multiplication. When we expand (2x + 3)(x + 5), we use the FOIL method (First, Outer, Inner, Last):

First: 2x * x = 2x²
Outer: 2x * 5 = 10x
Inner: 3 * x = 3x
Last: 3 * 5 = 15

Combining these terms, we get 2x² + 10x + 3x + 15 = 2x² + 13x + 15, which is our original expression. This demonstrates that (2x + 3)(x + 5) is indeed the correct factorization.

Applications of Factorization

The ability to factor quadratic expressions has significant applications across various mathematical fields:

Solving Quadratic Equations: Factoring is a crucial technique for solving quadratic equations. Once you've factored the quadratic expression, setting each factor equal to zero provides the solutions to the equation.
Simplifying Algebraic Expressions: Factorization simplifies complex algebraic expressions, making them easier to manipulate and understand.
Calculus: Factorization is essential in calculus for simplifying derivatives and integrals.
Graphing Quadratic Functions: The factored form of a quadratic expression reveals the x-intercepts (roots) of the corresponding quadratic function, which is vital for accurate graphing.

Frequently Asked Questions (FAQs)

Q1: What if I can't find the factors easily?

A1: If the trial-and-error method proves difficult, the AC method offers a more systematic approach. Alternatively, the quadratic formula always provides the roots, which can then be used to determine the factored form.

Q2: Can all quadratic expressions be factored?

A2: No. Some quadratic expressions cannot be factored using integers. In such cases, the quadratic formula is necessary to find the roots, and the expression might involve irrational or complex numbers.

Q3: Is there only one correct factorization?

A3: For a given quadratic expression, there's essentially only one unique factored form, although the order of the factors can be reversed. For example, (2x + 3)(x + 5) is equivalent to (x + 5)(2x + 3).

Q4: How can I check if my factorization is correct?

A4: Expand your factored expression using the FOIL method or distributive property. If you obtain the original quadratic expression, your factorization is correct.

Conclusion: Mastering Factorization

Factorization is a fundamental skill in algebra with far-reaching applications. While the initial learning curve might seem challenging, mastering various techniques like the AC method and understanding the underlying principles will significantly improve your algebraic skills. Remember that practice is key – the more you practice factoring, the quicker and more intuitive the process will become. By understanding the different approaches and utilizing them effectively, you'll gain confidence in tackling more complex algebraic problems and unlock a deeper appreciation for the beauty and power of mathematics. Don't be afraid to experiment with different methods and find the approach that works best for you. With consistent effort, you'll become a factorization expert in no time!