What Is The Factored Form Of 6x2 11x 10

What is the Factored Form of 6x² + 11x + 10? A Comprehensive Guide

Factoring quadratic expressions is a fundamental skill in algebra. Understanding how to factor allows you to solve quadratic equations, simplify expressions, and delve deeper into more advanced mathematical concepts. This comprehensive guide will explore the process of factoring the quadratic expression 6x² + 11x + 10, providing a step-by-step explanation, alternative methods, and practical applications.

Understanding Quadratic Expressions

Before we dive into factoring 6x² + 11x + 10, let's refresh our understanding of quadratic expressions. A quadratic expression is an algebraic expression of the form ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The highest power of the variable (x in this case) is 2, hence the term "quadratic."

Our specific expression, 6x² + 11x + 10, fits this form perfectly, with a = 6, b = 11, and c = 10. Factoring this expression means rewriting it as a product of two simpler expressions, typically two binomials.

Method 1: AC Method (Splitting the Middle Term)

The AC method, also known as the splitting the middle term method, is a widely used technique for factoring quadratic expressions. Here's how it works for 6x² + 11x + 10:

1. Find the product 'ac': Multiply the coefficient of the x² term (a) by the constant term (c): 6 * 10 = 60

2. Find two numbers that add up to 'b' and multiply to 'ac': We need two numbers that add up to 11 (the coefficient of the x term) and multiply to 60. These numbers are 6 and 5 (6 + 5 = 11 and 6 * 5 = 60).

3. Rewrite the middle term: Split the middle term (11x) using the two numbers we found: 6x² + 6x + 5x + 10

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

   • Group 1: 6x² + 6x = 6x(x + 1)
   • Group 2: 5x + 10 = 5(x + 1)

5. Factor out the common binomial: Notice that both groups have a common factor of (x + 1). Factor this out: (x + 1)(6x + 5)

Therefore, the factored form of 6x² + 11x + 10 is (x + 1)(6x + 5).

Method 2: Trial and Error

The trial and error method involves systematically testing different binomial pairs until you find the one that expands to the original quadratic expression. This method relies on understanding how binomial multiplication works. For 6x² + 11x + 10:

1. Consider the factors of the leading coefficient (6): The factors of 6 are (1, 6) and (2, 3).

2. Consider the factors of the constant term (10): The factors of 10 are (1, 10), (2, 5), (5,2) and (10,1).

3. Test different combinations: We need to find a combination that, when multiplied using the FOIL (First, Outer, Inner, Last) method, yields the original expression. Let's try a few:

   • (x + 1)(6x + 10): This gives 6x² + 16x + 10 (Incorrect)
   • (x + 2)(6x + 5): This gives 6x² + 17x + 10 (Incorrect)
   • (x + 5)(6x + 2): This gives 6x² + 32x + 10 (Incorrect)
   • (x + 1)(6x + 5): This gives 6x² + 11x + 10 (Correct!)

Again, we arrive at the factored form: (x + 1)(6x + 5). This method can be quicker with practice, but it may be less systematic than the AC method, particularly for more complex quadratics.

Method 3: Quadratic Formula (Indirect Factoring)

While not a direct factoring method, the quadratic formula can be used to find the roots (solutions) of the quadratic equation 6x² + 11x + 10 = 0. Once you have the roots, you can use them to construct the factored form.

The quadratic formula is: x = [-b ± √(b² - 4ac)] / 2a

For our equation: a = 6, b = 11, c = 10

x = [-11 ± √(11² - 4 * 6 * 10)] / (2 * 6) x = [-11 ± √(121 - 240)] / 12 x = [-11 ± √(-119)] / 12

Since the discriminant (b² - 4ac = -119) is negative, the roots are complex numbers. However, if we had real roots (r1 and r2), the factored form would be a(x - r1)(x - r2), where 'a' is the coefficient of x². The quadratic formula is extremely useful when factoring is difficult or impossible using other methods. It helps reveal underlying structure even when roots are not rational numbers.

Verifying the Factored Form

It's crucial to always verify your factored form by expanding it using the distributive property (FOIL):

(x + 1)(6x + 5) = x(6x) + x(5) + 1(6x) + 1(5) = 6x² + 5x + 6x + 5 = 6x² + 11x + 10

The expansion matches the original quadratic expression, confirming that (x + 1)(6x + 5) is the correct factored form.

Applications of Factoring Quadratic Expressions

Factoring quadratic expressions is not just an abstract algebraic exercise; it has numerous practical applications in various fields:

• Solving Quadratic Equations: Factoring is a key method for solving quadratic equations. Setting the factored expression equal to zero and solving for x provides the solutions to the equation.

• Graphing Quadratic Functions: The factored form reveals the x-intercepts (where the graph crosses the x-axis) of a parabola, making graphing easier.

• Calculus: Factoring is essential in calculus for simplifying expressions, finding derivatives and integrals, and solving optimization problems.

• Physics and Engineering: Quadratic equations and their solutions are frequently used in modeling projectile motion, oscillations, and other physical phenomena.

• Economics and Finance: Quadratic functions are used in economic models to describe things like profit maximization and cost minimization.

Further Exploration: Factoring More Complex Quadratic Expressions

While 6x² + 11x + 10 is a relatively straightforward quadratic to factor, the techniques discussed here can be extended to handle more complex expressions. These may involve:

• Leading coefficients that are not easily factored: Using the quadratic formula becomes even more valuable in these cases.

• Expressions with a greatest common factor (GCF): Always look for a GCF before attempting any other factoring methods. Factoring out the GCF simplifies the expression significantly.

• Difference of squares: Expressions of the form a² - b² can be factored as (a + b)(a - b).

• Perfect square trinomials: Expressions of the form a² + 2ab + b² or a² - 2ab + b² can be factored as (a + b)² or (a - b)², respectively.

Mastering these factoring techniques opens doors to solving more complex algebraic problems and lays the groundwork for success in higher-level mathematics and related fields. Remember that practice is key! The more you work with quadratic expressions, the more confident and efficient you'll become in factoring them.