Factoring Quadratics A Step By Step Guide To Factoring 6x² - 3x - 45
Factoring quadratic expressions is a fundamental skill in algebra. It allows us to simplify expressions, solve equations, and understand the behavior of polynomial functions. In this article, we will walk through the process of factoring the quadratic expression 6x² - 3x - 45 step-by-step, ensuring you understand each stage and the reasoning behind it. We will explore the common techniques used in factoring, paying particular attention to the extraction of the greatest common factor (GCF) and the application of the factoring by grouping method. Our goal is not just to arrive at the correct answer but to provide a comprehensive understanding that will enable you to tackle similar problems with confidence. By the end of this guide, you should be able to recognize common factors, set up the factoring process, and accurately determine the factored form of quadratic expressions.
1. Identifying the Greatest Common Factor (GCF)
When approaching a factoring problem, the first step should always be to look for the greatest common factor (GCF). The GCF is the largest factor that divides evenly into all terms of the expression. Identifying and factoring out the GCF simplifies the expression, making subsequent factoring steps easier. In the quadratic expression 6x² - 3x - 45, we need to examine the coefficients (6, -3, and -45) to find the largest number that divides all three. The factors of 6 are 1, 2, 3, and 6. The factors of 3 are 1 and 3. The factors of 45 are 1, 3, 5, 9, 15, and 45. By comparing these, we can see that the greatest common factor of 6, 3, and 45 is 3. Now, we factor out the 3 from the original expression:
6x² - 3x - 45 = 3(2x² - x - 15)
Factoring out the GCF of 3 reduces the complexity of the quadratic expression inside the parentheses, making it easier to factor further. This step is crucial because it not only simplifies the expression but also ensures that we obtain the completely factored form. Overlooking the GCF can lead to incorrect factoring or leaving the expression in a partially factored state. In our case, factoring out the 3 transforms the initial quadratic into a more manageable form, which we can now proceed to factor by looking at the trinomial 2x² - x - 15.
2. Factoring the Trinomial: 2x² - x - 15
After extracting the GCF, we are left with the trinomial 2x² - x - 15. Factoring this trinomial involves breaking it down into two binomials. There are several techniques to accomplish this, such as the trial-and-error method or the factoring by grouping method. We will employ the factoring by grouping method, which is a systematic approach that ensures accuracy. This method involves finding two numbers that satisfy two conditions: they must multiply to the product of the leading coefficient (2) and the constant term (-15), and they must add up to the middle coefficient (-1). So, we are looking for two numbers that multiply to 2 * -15 = -30 and add to -1.
Let’s list the pairs of factors of -30:
- 1 and -30
- -1 and 30
- 2 and -15
- -2 and 15
- 3 and -10
- -3 and 10
- 5 and -6
- -5 and 6
Among these pairs, the pair that adds up to -1 is 5 and -6. Now, we rewrite the middle term (-x) using these two numbers:
2x² - x - 15 = 2x² + 5x - 6x - 15
Next, we group the terms in pairs:
(2x² + 5x) + (-6x - 15)
We then factor out the GCF from each pair. From the first group, the GCF is x:
x(2x + 5)
From the second group, the GCF is -3:
-3(2x + 5)
So, the expression becomes:
x(2x + 5) - 3(2x + 5)
Notice that we now have a common binomial factor, (2x + 5). We factor out this common binomial:
(2x + 5)(x - 3)
Thus, the factored form of the trinomial 2x² - x - 15 is (2x + 5)(x - 3). This decomposition is a critical step in solving quadratic equations and simplifying algebraic expressions. Understanding this process allows us to handle more complex factoring problems effectively.
3. Combining the GCF and the Factored Trinomial
Having factored the trinomial 2x² - x - 15 into (2x + 5)(x - 3) and recalling the GCF we factored out earlier, which was 3, we now combine these results to obtain the complete factorization of the original quadratic expression, 6x² - 3x - 45. The process is straightforward: we simply multiply the GCF by the factored form of the trinomial. This step ensures that we have accounted for all factors of the original expression and have expressed it in its simplest factored form.
We start by writing down the factored trinomial:
(2x + 5)(x - 3)
Then, we include the GCF that we factored out in the first step:
3(2x + 5)(x - 3)
This combined expression, 3(2x + 5)(x - 3), represents the fully factored form of the original quadratic expression 6x² - 3x - 45. This means that if we were to expand this expression by multiplying out the terms, we would arrive back at the original quadratic expression. Combining the GCF with the factored trinomial is a crucial step in completing the factoring process. It ensures that the final answer is in the most simplified form and that all factors have been accounted for. This comprehensive factorization is essential for solving quadratic equations, simplifying rational expressions, and various other algebraic manipulations.
4. Verifying the Solution
To ensure the accuracy of our factoring, it's crucial to verify the solution. Verification involves expanding the factored expression to see if it matches the original quadratic expression. This step helps catch any errors made during the factoring process. We will expand the expression 3(2x + 5)(x - 3) to confirm that it equals 6x² - 3x - 45.
First, let’s expand the two binomials (2x + 5)(x - 3):
(2x + 5)(x - 3) = 2x(x - 3) + 5(x - 3)
Distribute 2x and 5 across the terms in the parentheses:
= 2x² - 6x + 5x - 15
Combine like terms:
= 2x² - x - 15
Now, we multiply the resulting trinomial by the GCF, which is 3:
3(2x² - x - 15) = 3(2x²) - 3(x) - 3(15)
Distribute the 3 across each term:
= 6x² - 3x - 45
The expanded expression 6x² - 3x - 45 matches the original quadratic expression. This confirms that our factoring is correct. Verification is an essential step in the factoring process, as it ensures that the factored form is equivalent to the original expression. By expanding the factored expression and comparing it to the original, we can be confident in the accuracy of our solution.
5. Conclusion
In conclusion, we have successfully factored the quadratic expression 6x² - 3x - 45. The process involved several key steps: first, identifying and factoring out the greatest common factor (GCF), which was 3. This simplified the expression to 3(2x² - x - 15). Next, we factored the trinomial 2x² - x - 15 using the factoring by grouping method, which resulted in (2x + 5)(x - 3). Combining the GCF with the factored trinomial, we obtained the complete factorization: 3(2x + 5)(x - 3). Finally, we verified our solution by expanding the factored expression and confirming that it matched the original quadratic expression.
Factoring quadratic expressions is a crucial skill in algebra, with applications in solving equations, simplifying expressions, and understanding the behavior of polynomial functions. The ability to identify the GCF and apply techniques such as factoring by grouping is essential for mastering this skill. By following a systematic approach and verifying the results, you can confidently factor quadratic expressions and apply this knowledge to more complex problems in algebra and beyond. The correct answer to the factorization of 6x² - 3x - 45 is 3(2x + 5)(x - 3).
