Factoring The Quadratic Expression 10y² - 11y - 6

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Factoring quadratic expressions is a fundamental skill in algebra, crucial for solving equations, simplifying expressions, and understanding the behavior of polynomial functions. In this comprehensive guide, we will delve into the process of factoring the quadratic expression 10y² - 11y - 6. We will explore different methods, provide step-by-step explanations, and offer examples to ensure a clear understanding. This article aims to equip you with the knowledge and confidence to tackle similar factoring problems effectively.

Understanding Quadratic Expressions

Before we dive into the specifics of factoring 10y² - 11y - 6, let's establish a solid foundation by understanding the general form of quadratic expressions. A quadratic expression is a polynomial of degree two, meaning the highest power of the variable is two. The standard form of a quadratic expression is:

ax² + bx + c

Where:

  • a, b, and c are constants (numbers).
  • x is the variable.

In our case, the quadratic expression 10y² - 11y - 6 fits this form perfectly. Here, a = 10, b = -11, and c = -6. Recognizing these coefficients is the first step in the factoring process.

The goal of factoring is to rewrite the quadratic expression as a product of two binomials. A binomial is a polynomial with two terms. For example, (2y + 1) and (5y - 6) are binomials. When we multiply these binomials together, we should obtain the original quadratic expression, 10y² - 11y - 6.

Methods for Factoring Quadratic Expressions

There are several methods for factoring quadratic expressions, but we will focus on two common and effective techniques:

  1. The Trial and Error Method (also known as the Guess and Check Method)
  2. The AC Method (also known as the Grouping Method)

We will explore both methods in detail, demonstrating their application to factoring 10y² - 11y - 6.

1. The Trial and Error Method

The trial and error method involves systematically guessing and checking different combinations of binomials until we find the pair that multiplies to give the original quadratic expression. This method relies on understanding how binomial multiplication works, particularly the FOIL method (First, Outer, Inner, Last).

Let's apply the trial and error method to factor 10y² - 11y - 6:

Step 1: Identify Possible Factors of 'a' and 'c'

In our expression, a = 10 and c = -6. We need to find pairs of factors for both 10 and -6.

  • Factors of 10: (1, 10), (2, 5)
  • Factors of -6: (1, -6), (-1, 6), (2, -3), (-2, 3)

Step 2: Create Potential Binomial Pairs

We will now create potential binomial pairs using these factors. Remember that the first terms of the binomials must multiply to give 10y², and the last terms must multiply to give -6. Here are some possible pairs:

  • (y + 1)(10y - 6)
  • (y - 1)(10y + 6)
  • (2y + 1)(5y - 6)
  • (2y - 1)(5y + 6)
  • (2y + 2)(5y - 3)
  • (2y - 2)(5y + 3)
  • (2y + 3)(5y - 2)
  • (2y - 3)(5y + 2)

Step 3: Test the Binomial Pairs using the FOIL Method

We will now use the FOIL method to multiply each binomial pair and see if it matches our original expression, 10y² - 11y - 6.

Let's start with (2y + 1)(5y - 6):

  • First: 2y * 5y = 10y²
  • Outer: 2y * -6 = -12y
  • Inner: 1 * 5y = 5y
  • Last: 1 * -6 = -6

Combining these terms, we get:

10y² - 12y + 5y - 6 = 10y² - 7y - 6

This does not match our original expression, so we move on to the next pair.

Let's try (2y - 3)(5y + 2):

  • First: 2y * 5y = 10y²
  • Outer: 2y * 2 = 4y
  • Inner: -3 * 5y = -15y
  • Last: -3 * 2 = -6

Combining these terms, we get:

10y² + 4y - 15y - 6 = 10y² - 11y - 6

This matches our original expression! Therefore, the factored form of 10y² - 11y - 6 is (2y - 3)(5y + 2).

2. The AC Method

The AC method provides a more systematic approach to factoring quadratic expressions, especially when the coefficient of the squared term (a) is not equal to 1. This method involves finding two numbers that satisfy specific conditions related to the coefficients a, b, and c.

Let's use the AC method to factor 10y² - 11y - 6:

Step 1: Multiply 'a' and 'c'

In our expression, a = 10 and c = -6. Multiply these values:

10 * -6 = -60

Step 2: Find Two Numbers That Multiply to 'ac' and Add Up to 'b'

We need to find two numbers that multiply to -60 and add up to b = -11. Let's list the factor pairs of -60 and check their sums:

  • 1 and -60 (sum: -59)
  • -1 and 60 (sum: 59)
  • 2 and -30 (sum: -28)
  • -2 and 30 (sum: 28)
  • 3 and -20 (sum: -17)
  • -3 and 20 (sum: 17)
  • 4 and -15 (sum: -11) <- This is the pair we need!
  • -4 and 15 (sum: 11)
  • 5 and -12 (sum: -7)
  • -5 and 12 (sum: 7)
  • 6 and -10 (sum: -4)
  • -6 and 10 (sum: 4)

The numbers 4 and -15 satisfy our conditions: 4 * -15 = -60 and 4 + (-15) = -11.

Step 3: Rewrite the Middle Term Using the Two Numbers

We will now rewrite the middle term, -11y, using the numbers 4 and -15:

10y² - 11y - 6 = 10y² + 4y - 15y - 6

Step 4: Factor by Grouping

We will now group the first two terms and the last two terms and factor out the greatest common factor (GCF) from each group:

(10y² + 4y) + (-15y - 6)

From the first group, the GCF is 2y:

2y(5y + 2)

From the second group, the GCF is -3:

-3(5y + 2)

Now we have:

2y(5y + 2) - 3(5y + 2)

Notice that both terms have a common factor of (5y + 2). We can factor this out:

(5y + 2)(2y - 3)

Therefore, the factored form of 10y² - 11y - 6 is (2y - 3)(5y + 2), which matches the result we obtained using the trial and error method.

Conclusion

In this comprehensive guide, we have successfully factored the quadratic expression 10y² - 11y - 6 using two different methods: the trial and error method and the AC method. Both methods are valuable tools in your algebraic arsenal. The trial and error method can be quicker for simpler expressions, while the AC method provides a more systematic approach for complex expressions.

The key to mastering factoring is practice. Work through various examples, and you will develop a strong intuition for identifying the correct factors. Remember to always double-check your answer by multiplying the binomials to ensure they equal the original quadratic expression.

Factoring quadratic expressions is a critical skill that opens the door to solving more advanced algebraic problems. By understanding the concepts and methods presented in this guide, you will be well-equipped to tackle a wide range of factoring challenges.

This article has provided a detailed explanation of how to factor the quadratic expression 10y² - 11y - 6. We explored the fundamental concepts of quadratic expressions, delved into two effective factoring methods, and provided step-by-step instructions with clear examples. With consistent practice, you can confidently factor quadratic expressions and excel in your algebraic endeavors.