Factoring Quadratic Expressions: A Step-by-Step Guide with 6x^2 - 5x - 6

In mathematics, factoring quadratic expressions is a fundamental concept, essential for solving polynomial equations and simplifying algebraic expressions. This article provides a detailed, step-by-step guide on how to factor the quadratic expression 6x^2 - 5x - 6. Understanding these methods is crucial for students and professionals alike, as it forms the basis for more complex mathematical problem-solving.

What is Factoring?

Factoring, in the context of algebra, refers to the process of breaking down a polynomial into a product of simpler expressions. The aim is to find the linear factors of a quadratic expression or polynomial. The method of factoring by grouping is particularly suitable for expressions where the coefficient of (x^2) is not 1.

Step 1: Identify the Coefficients

To begin, let us identify the coefficients of the quadratic expression (6x^2 - 5x - 6).

a: The coefficient of the (x^2) term, which is 6. b: The coefficient of the (x) term, which is -5. c: The constant term, which is -6.

Step 2: Calculate the Product ac

The next step involves calculating the product of the coefficients (a) and (c). This product will help us find two numbers that fit specific criteria.

Calculating (ac)

Here, (a 6) and (c -6). The product (ac) is:

[ac 6 times -6 -36]

Step 3: Find Two Numbers That Multiply to ac and Add to b

In this step, we need to find two numbers that multiply to (-36) and add to (-5). These numbers will help us split the middle term of the quadratic expression.

After some trial and error, we find that 4 and -9 are the numbers we require because: 4 (times) -9 -36 4 (-9) -5

Step 4: Rewrite the Middle Term Using These Numbers

Now that we have these numbers, we can rewrite the expression (-5x) as (4x - 9x).

[6x^2 - 5x - 6 6x^2 4x - 9x - 6]

Step 5: Group the Terms

We group the first two terms and the last two terms:

[6x^2 4x - 9x - 6 (6x^2 4x) (-9x - 6)]

Step 6: Factor Out the Common Factors in Each Group

In this step, we factor out the common factor from each group:

From (6x^2 4x), we factor out 2x: From (-9x - 6), we factor out -3: [2x(3x 2) - 3(3x 2)]

Step 7: Factor Out the Common Binomial Factor

Notice that (3x 2) is a common factor in both groups. We can factor this out:

[2x(3x 2) - 3(3x 2) (3x 2)(2x - 3)]

Final Result

The completely factored form of the quadratic expression (6x^2 - 5x - 6) is:

[boxed{(3x 2)(2x - 3)}]

Conclusion

Factoring the quadratic expression (6x^2 - 5x - 6) involves the method of factoring by grouping. By following these seven steps, we can break down the expression into its simpler factors. Understanding this process not only aids in solving polynomial equations but also enhances overall mathematical proficiency.

For more resources and practice problems, consider exploring additional articles and tutorials on factoring quadratic equations. Practice is key to mastering this essential math skill.