Solving Quadratic Equations: A Step-by-Step Guide For 12a²

Introduction

In this comprehensive guide, we'll dive into the process of solving the quadratic equation 12a² - 22a - 5 = -5a. Quadratic equations, characterized by their highest power of 2, frequently appear in various mathematical and scientific contexts. Mastering the techniques to solve them is an essential skill. In this article, we will walk through each step in detail, ensuring a clear and thorough understanding. Whether you're a student tackling algebra or simply brushing up on your math skills, this guide will provide you with the tools you need to confidently solve similar problems. We will cover the initial setup, simplification, factoring (if possible), and the application of the quadratic formula, ensuring you grasp every aspect of the solution. Let’s begin by understanding the basics of quadratic equations and then proceed to tackle the problem at hand. Remember, the key is to break down the problem into manageable steps, and that’s exactly what we’re going to do here. So, grab your pen and paper, and let’s get started!

1. Understanding Quadratic Equations

Before we tackle the specific equation, let's briefly discuss what quadratic equations are. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable. The term 'a' cannot be zero, as that would make it a linear equation. Recognizing this standard form is the first step in solving any quadratic equation. The solutions to a quadratic equation are also known as its roots or zeros, which are the values of 'x' that satisfy the equation. These roots can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding these methods and when to use them is crucial for effectively solving quadratic equations. In this case, our equation involves the variable 'a' instead of 'x', but the principles remain the same. So, with this foundational knowledge in mind, we can move forward to applying these concepts to our specific problem. Remember, the goal is to transform the given equation into the standard form so that we can easily apply the appropriate solving techniques. This understanding will guide us as we proceed with the step-by-step solution.

2. Simplifying the Equation

Our initial equation is 12a² - 22a - 5 = -5a. The first step in solving any equation is to simplify it and bring it to the standard quadratic form. This involves moving all terms to one side of the equation, leaving zero on the other side. To do this, we need to add 5a to both sides of the equation. This will eliminate the -5a on the right side and combine like terms on the left side. Adding 5a to both sides, we get: 12a² - 22a - 5 + 5a = -5a + 5a. Simplifying this, we combine the '-22a' and '+5a' terms, which gives us -17a. Therefore, the equation becomes: 12a² - 17a - 5 = 0. Now, we have the equation in the standard quadratic form: ax² + bx + c = 0, where a = 12, b = -17, and c = -5. This simplified form is crucial because it allows us to easily identify the coefficients needed for factoring or applying the quadratic formula. With the equation now in this standard form, we can proceed to the next step, which involves determining the most appropriate method for finding the solutions. Whether we choose to factor, complete the square, or use the quadratic formula will depend on the specific characteristics of the equation.

3. Attempting to Factor the Quadratic

Now that we have our simplified equation, 12a² - 17a - 5 = 0, let's see if we can solve it by factoring. Factoring involves breaking down the quadratic expression into two binomials. If we can factor the equation, it often provides a quicker solution than using the quadratic formula. To factor, we need to find two numbers that multiply to give the product of the leading coefficient (12) and the constant term (-5), which is -60, and add up to the middle coefficient (-17). We are looking for two numbers that satisfy these conditions. After considering different pairs of factors of -60, we find that the numbers -20 and 3 fit the criteria. -20 multiplied by 3 equals -60, and -20 plus 3 equals -17. Now, we rewrite the middle term (-17a) using these two numbers: 12a² - 20a + 3a - 5 = 0. Next, we factor by grouping. We group the first two terms and the last two terms: (12a² - 20a) + (3a - 5) = 0. We factor out the greatest common factor from each group. From the first group, we can factor out 4a, and from the second group, we can factor out 1: 4a(3a - 5) + 1(3a - 5) = 0. Notice that we now have a common binomial factor, which is (3a - 5). We factor this out: (3a - 5)(4a + 1) = 0. Now that we have factored the quadratic equation, we can proceed to find the values of 'a' that make the equation true. If factoring isn’t immediately obvious, remember it might take a few tries or exploring different factor pairs. If factoring proves too difficult, the quadratic formula is always a reliable alternative. In our case, we successfully factored the equation, which simplifies the process of finding the solutions.

4. Solving for 'a' Using the Factored Form

Having successfully factored the quadratic equation into (3a - 5)(4a + 1) = 0, we can now solve for 'a'. The principle we use here is the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero and solve for 'a'. First, we set the factor (3a - 5) equal to zero: 3a - 5 = 0. To solve for 'a', we add 5 to both sides: 3a = 5. Then, we divide both sides by 3: a = 5/3. This gives us our first solution for 'a'. Next, we set the factor (4a + 1) equal to zero: 4a + 1 = 0. To solve for 'a', we subtract 1 from both sides: 4a = -1. Then, we divide both sides by 4: a = -1/4. This gives us our second solution for 'a'. Therefore, the solutions to the quadratic equation 12a² - 17a - 5 = 0 are a = 5/3 and a = -1/4. These are the values of 'a' that make the original equation true. By factoring the quadratic equation and applying the zero-product property, we efficiently found the solutions. It's always a good practice to check these solutions by substituting them back into the original equation to ensure they satisfy it. This confirms the accuracy of our work and provides confidence in our answers.

5. Verification of Solutions

To ensure the accuracy of our solutions, it's crucial to verify them. We do this by substituting each value of 'a' that we found back into the original equation, 12a² - 22a - 5 = -5a, and checking if the equation holds true. Let's start with the first solution, a = 5/3. We substitute a = 5/3 into the equation: 12(5/3)² - 22(5/3) - 5 = -5(5/3). Now, we simplify each term: 12(25/9) - 110/3 - 5 = -25/3. This simplifies to: 300/9 - 110/3 - 5 = -25/3. To combine these terms, we need a common denominator, which is 9: 300/9 - 330/9 - 45/9 = -25/3. Combining the terms on the left side gives us: -75/9 = -25/3. Simplifying -75/9, we get -25/3, which matches the right side of the equation. Thus, a = 5/3 is indeed a valid solution. Now, let's verify the second solution, a = -1/4. We substitute a = -1/4 into the equation: 12(-1/4)² - 22(-1/4) - 5 = -5(-1/4). Simplifying each term: 12(1/16) + 22/4 - 5 = 5/4. This simplifies to: 12/16 + 22/4 - 5 = 5/4. To combine these terms, we need a common denominator, which is 16: 12/16 + 88/16 - 80/16 = 5/4. Combining the terms on the left side gives us: 20/16 = 5/4. Simplifying 20/16, we get 5/4, which matches the right side of the equation. Thus, a = -1/4 is also a valid solution. By verifying both solutions, we can be confident that our answers are correct. This step is a critical part of the problem-solving process, as it helps to catch any errors that may have occurred during the earlier steps.

Conclusion

In this guide, we've walked through the process of solving the quadratic equation 12a² - 22a - 5 = -5a. We started by simplifying the equation into standard quadratic form, then we factored the quadratic expression, and finally, we used the zero-product property to find the solutions. We also verified our solutions to ensure their accuracy. The solutions to the equation are a = 5/3 and a = -1/4. Understanding these steps provides a solid foundation for solving other quadratic equations. Quadratic equations are a fundamental topic in algebra, and mastering the techniques to solve them is essential for further mathematical studies. Remember, the key is to break down the problem into manageable steps and to practice regularly. By following the steps outlined in this guide, you can confidently tackle a wide range of quadratic equations. Keep practicing, and you'll find that solving these equations becomes more intuitive over time. For additional resources and practice problems on quadratic equations, you may want to visit Khan Academy's Algebra Section.