Factor 6x 2 13x 5

Factoring the Quadratic Expression 6x² + 13x + 5

Factoring quadratic expressions is a fundamental skill in algebra. It's a process that allows us to rewrite a quadratic expression as a product of two simpler expressions, making it easier to solve equations, simplify expressions, and analyze graphs. This article will guide you through the process of factoring the specific quadratic expression 6x² + 13x + 5, explaining the methods involved and providing a deep understanding of the underlying principles. We'll explore various techniques and address common questions, ensuring you gain confidence in tackling similar problems.

Understanding Quadratic Expressions

Before diving into the factoring process, let's briefly review what a quadratic expression is. 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 (numbers). In our case, a = 6, b = 13, and c = 5.

Method 1: AC Method (Factoring by Grouping)

The AC method, also known as factoring by grouping, is a systematic approach to factoring quadratic expressions. It involves finding two numbers that satisfy specific conditions related to the coefficients a and c.

Steps:

1. Find the product AC: Multiply the coefficient of the x² term (a) by the constant term (c). In our example, AC = 6 * 5 = 30.

2. 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 the x term (b), which is 13, and multiply to 30 (AC). These numbers are 3 and 10 (3 + 10 = 13 and 3 * 10 = 30).

3. Rewrite the expression: Rewrite the middle term (13x) as the sum of the two numbers we found, multiplied by x. This gives us: 6x² + 3x + 10x + 5.

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

   3x(2x + 1) + 5(2x + 1)

5. Factor out the common binomial: Notice that (2x + 1) is a common factor in both terms. Factor it out:

   (2x + 1)(3x + 5)

Therefore, the factored form of 6x² + 13x + 5 is (2x + 1)(3x + 5).

Method 2: Trial and Error

This method involves directly trying different combinations of factors of a and c until you find the pair that produces the correct middle term (b). While less systematic than the AC method, it can be faster for simpler quadratics.

Steps:

1. Find factors of a and c: The factors of a (6) are 1, 2, 3, and 6. The factors of c (5) are 1 and 5.

2. Test combinations: We need to find combinations of these factors that, when multiplied and added, yield 13x. Let's try some combinations:

   • (2x + 1)(3x + 5): This gives 6x² + 10x + 3x + 5 = 6x² + 13x + 5. This works!

   • (2x + 5)(3x + 1): This gives 6x² + x + 15x + 5 = 6x² + 16x + 5. This doesn't work.

   • (x+1)(6x+5): This gives 6x² + 11x +5. This doesn't work either.

   • and so on...

After testing a few combinations, we find that (2x + 1)(3x + 5) is the correct factorization. The trial-and-error method becomes less efficient as the coefficients become larger and have more factors.

Checking Your Answer

It's always a good idea to check your answer by expanding the factored expression. Multiplying (2x + 1) and (3x + 5) using the FOIL method (First, Outer, Inner, Last) gives:

(2x * 3x) + (2x * 5) + (1 * 3x) + (1 * 5) = 6x² + 10x + 3x + 5 = 6x² + 13x + 5

This confirms that our factorization is correct.

The Significance of Factoring

Factoring quadratic expressions is more than just a mathematical exercise; it has significant applications in various areas of mathematics and beyond:

• Solving Quadratic Equations: Once a quadratic expression is factored, it becomes easier to solve the corresponding quadratic equation (e.g., 6x² + 13x + 5 = 0). Setting each factor to zero allows you to find the roots (solutions) of the equation.

• Simplifying Expressions: Factoring can simplify complex algebraic expressions, making them easier to manipulate and understand.

• Graphing Quadratic Functions: The factored form of a quadratic expression reveals the x-intercepts (where the graph crosses the x-axis) of the corresponding quadratic function.

• Calculus: Factoring plays a crucial role in calculus, particularly in finding derivatives and integrals.

Frequently Asked Questions (FAQ)

Q1: What if the quadratic expression cannot be factored easily?

A1: Not all quadratic expressions can be factored using integer coefficients. In such cases, you can use the quadratic formula to find the roots, or you might need to use more advanced techniques like completing the square.

Q2: Is there only one correct way to factor a quadratic expression?

A2: No, there isn't. While the factored form might look different depending on the order of the factors (e.g., (2x + 1)(3x + 5) is the same as (3x + 5)(2x + 1)), the underlying factorization remains the same.

Q3: What happens if the coefficient of x² (a) is 1?

A3: Factoring becomes simpler. You just need to find two numbers that add up to b and multiply to c.

Q4: What if the quadratic expression has a greatest common factor (GCF)?

A4: It's always a good practice to factor out the GCF first. For example, if the expression was 12x² + 26x + 10, you would first factor out 2 to get 2(6x² + 13x + 5), and then factor the quadratic inside the parentheses using the methods described above.

Conclusion

Factoring the quadratic expression 6x² + 13x + 5, as demonstrated using the AC method and trial-and-error, provides a solid foundation for understanding the broader concept of factoring quadratic expressions. Mastering this skill is crucial for success in algebra and subsequent mathematical studies. Remember to practice regularly, explore different methods, and always check your answers to build confidence and proficiency. The ability to factor quadratics opens doors to a deeper understanding of algebraic manipulation and its applications in diverse fields. By understanding the underlying principles and applying the techniques outlined, you'll be well-equipped to tackle a wide range of quadratic expressions with ease and confidence.