Solving Quadratic Equations: Step-by-Step Guide
Hey math enthusiasts! Ever stumbled upon a quadratic equation and thought, "Whoa, where do I even begin?" Well, fear not! Today, we're diving headfirst into the world of solving quadratic equations, specifically tackling the equation 10y² + 7y = 12. We'll break down the process step-by-step, making sure you grasp every detail. This guide is designed to be your go-to resource, whether you're brushing up on your algebra skills or learning this concept for the first time. We'll explore different methods, ensuring you have a solid understanding of how to find the solutions (also known as roots) of these types of equations. Get ready to flex those math muscles and unlock the secrets behind quadratic equations. Let's get started!
Understanding Quadratic Equations
Before we jump into the nitty-gritty of solving 10y² + 7y = 12, let's take a quick look at what quadratic equations are all about. In simple terms, a quadratic equation is an equation that can be written in the form ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The highest power of the variable (in this case, x or y) is 2, hence the name "quadratic" (from the Latin word "quadratus," meaning square). These equations often pop up in various fields, from physics and engineering to economics and even computer science. Recognizing a quadratic equation is the first step toward solving it. The general form highlights three key components: the quadratic term (ax²), the linear term (bx), and the constant term (c). The presence of the x² term is what makes it a quadratic equation, and the coefficients a, b, and c determine the shape and position of the parabola when graphed. Different methods exist for solving quadratic equations, each with its own advantages and when it's appropriate to apply it. These methods include factoring, completing the square, and using the quadratic formula. The choice of which method to use often depends on the specific equation and personal preference. The goal, however, always remains the same: to find the values of x or y that satisfy the equation. These values, known as the roots or solutions, represent the points where the parabola crosses the x-axis, providing critical information about the behavior of the quadratic function.
Key Components of a Quadratic Equation
Let's break down the components of a quadratic equation using our example, 10y² + 7y = 12, and transform it into the standard form. First, we need to rearrange the equation so that it equals zero. Subtracting 12 from both sides, we get 10y² + 7y - 12 = 0. Now, we can identify the coefficients: a = 10, b = 7, and c = -12. a is the coefficient of the y² term, b is the coefficient of the y term, and c is the constant term. These coefficients are crucial as they determine the specific solution methods we can use, and understanding their role is fundamental. Recognizing these components is the first step toward finding the roots of the equation. Each coefficient plays a role in the equation's behavior, influencing the graph of the function and the points where it intersects the x-axis. This understanding will become more critical as we move forward and start solving our equation using different approaches. The values of a, b, and c also determine whether the solutions will be real numbers, complex numbers, or if there will be multiple or no solutions at all. The sign of the coefficients also tells us about the direction and position of the parabola's graph. A positive a means the parabola opens upwards, while a negative a means it opens downwards. This knowledge is not only important for solving the equation, but also provides insight into the nature of the function represented.
Method 1: Factoring to Solve the Equation
Alright, let's start solving 10y² + 7y - 12 = 0 using factoring. Factoring involves breaking down the quadratic expression into the product of two binomials. This method works well when the quadratic expression can be easily factored, which is not always the case, but it's a great place to start! The goal here is to find two numbers that multiply to give the product of a and c (which is 10 * -12 = -120) and add up to b (which is 7). These numbers are 15 and -8 because 15 * -8 = -120 and 15 + (-8) = 7. Next, we rewrite the middle term, 7y, using these two numbers. So, 10y² + 7y - 12 = 0 becomes 10y² + 15y - 8y - 12 = 0. Now, we group the terms and factor by grouping. Group the first two terms and the last two terms: (10y² + 15y) + (-8y - 12) = 0. Factor out the greatest common factor (GCF) from each group. From the first group, we can factor out 5y, and from the second group, we can factor out -4, so it becomes 5y(2y + 3) - 4(2y + 3) = 0. Notice that we now have a common binomial factor, (2y + 3). Factor out (2y + 3): (2y + 3)(5y - 4) = 0. Finally, set each factor equal to zero and solve for y. For the first factor, 2y + 3 = 0, so 2y = -3, and y = -3/2 or -1.5. For the second factor, 5y - 4 = 0, so 5y = 4, and y = 4/5 or 0.8. Thus, the solutions to the equation are y = -1.5 and y = 0.8. Factoring is often the quickest method when it's applicable and helps build a strong algebraic understanding. Mastering this approach can significantly speed up your problem-solving skills, and recognizing when an expression can be factored quickly will save you time and effort. Remember, not all quadratic equations are easy to factor. In such cases, other methods like the quadratic formula will become your best friends.
Step-by-Step Factoring Process
Let's recap the steps involved in factoring the equation 10y² + 7y - 12 = 0. First, identify a, b, and c. In this case, a = 10, b = 7, and c = -12. Multiply a and c: 10 * -12 = -120. Find two numbers that multiply to -120 and add to 7. These numbers are 15 and -8. Rewrite the middle term using these two numbers: 10y² + 15y - 8y - 12 = 0. Group the terms and factor by grouping: (10y² + 15y) + (-8y - 12) = 0. Factor out the GCF from each group: 5y(2y + 3) - 4(2y + 3) = 0. Factor out the common binomial factor: (2y + 3)(5y - 4) = 0. Set each factor equal to zero and solve for y: 2y + 3 = 0, which gives y = -1.5, and 5y - 4 = 0, which gives y = 0.8. The solutions are y = -1.5 and y = 0.8. Remember, practice makes perfect. The more you work through examples, the more proficient you'll become at recognizing patterns and efficiently factoring quadratic equations.
Method 2: Using the Quadratic Formula
If factoring seems tricky or the numbers aren't cooperating, the quadratic formula is your reliable backup. This formula is a universal tool that works for any quadratic equation in the form ax² + bx + c = 0. The quadratic formula is y = (-b ± √(b² - 4ac)) / 2a. Let's plug in the values from our equation, 10y² + 7y - 12 = 0. We already know that a = 10, b = 7, and c = -12. Substitute these values into the formula: y = (-7 ± √(7² - 4 * 10 * -12)) / (2 * 10). Simplify the expression inside the square root: 7² - 4 * 10 * -12 = 49 + 480 = 529. So, the formula becomes y = (-7 ± √529) / 20. The square root of 529 is 23, so we have y = (-7 ± 23) / 20. Now we have two possible solutions: y = (-7 + 23) / 20 = 16 / 20 = 4/5 or 0.8, and y = (-7 - 23) / 20 = -30 / 20 = -3/2 or -1.5. These are the same solutions we found using factoring, which validates our results. The quadratic formula is powerful because it always gives you the correct answer, regardless of how complex the equation looks, and is especially helpful when dealing with non-factorable quadratics. Understanding this formula is a fundamental skill in algebra, enabling you to solve a wide variety of problems. Make sure to keep this formula close by!
Applying the Quadratic Formula Step-by-Step
Let's go through the quadratic formula step by step. First, identify a, b, and c from the standard form ax² + bx + c = 0. In our equation, 10y² + 7y - 12 = 0, we have a = 10, b = 7, and c = -12. Write down the quadratic formula: y = (-b ± √(b² - 4ac)) / 2a. Substitute the values of a, b, and c into the formula: y = (-7 ± √(7² - 4 * 10 * -12)) / (2 * 10). Simplify the expression under the square root: b² - 4ac = 7² - 4 * 10 * -12 = 49 + 480 = 529. Calculate the square root: √529 = 23. Calculate the two possible values of y: y = (-7 + 23) / 20 and y = (-7 - 23) / 20. Simplify to find the solutions: y = 16 / 20 = 0.8 and y = -30 / 20 = -1.5. The solutions are y = 0.8 and y = -1.5. Using this formula is a systematic approach that guarantees results, even when factoring is challenging, making it an indispensable tool for every algebra student. This method ensures you can solve any quadratic equation with confidence, making it a cornerstone for higher-level mathematics.
Method 3: Completing the Square
Completing the square is another method to solve quadratic equations. This technique involves manipulating the equation to create a perfect square trinomial. While it can be more involved than factoring or using the quadratic formula, it is a great method to learn and understand the structure of quadratic equations more deeply. Let's start with our equation 10y² + 7y - 12 = 0. First, divide the equation by a (in this case, 10) to make the coefficient of the y² term equal to 1. This gives us y² + (7/10)y - 12/10 = 0, which simplifies to y² + 0.7y - 1.2 = 0. Move the constant term to the right side of the equation: y² + 0.7y = 1.2. Next, take half of the coefficient of the y term (0.7), square it ((0.7 / 2)² = 0.35² = 0.1225), and add it to both sides of the equation: y² + 0.7y + 0.1225 = 1.2 + 0.1225. This simplifies to (y + 0.35)² = 1.3225. Now, take the square root of both sides: y + 0.35 = ±√1.3225. The square root of 1.3225 is approximately 1.15. So, y + 0.35 = ±1.15. Solve for y: y = -0.35 ± 1.15. Calculate the two solutions: y = -0.35 + 1.15 = 0.8 and y = -0.35 - 1.15 = -1.5. Hence, the solutions are y = 0.8 and y = -1.5, which again matches the result we found using the other methods. Completing the square is very useful as it provides a deep understanding of the structure of quadratic equations and is the basis of the quadratic formula.
The Process of Completing the Square
Let's walk through the steps of completing the square for our equation. First, write the equation in the form ax² + bx + c = 0. If a ≠1, divide the entire equation by a. For 10y² + 7y - 12 = 0, divide by 10 to get y² + 0.7y - 1.2 = 0. Move the constant term to the right side: y² + 0.7y = 1.2. Calculate (b/2)², in this case (0.7/2)² = 0.1225. Add this value to both sides of the equation: y² + 0.7y + 0.1225 = 1.2 + 0.1225. Simplify the left side to a perfect square: (y + 0.35)² = 1.3225. Take the square root of both sides: y + 0.35 = ±1.15. Solve for y: y = -0.35 ± 1.15. Find the two solutions: y = 0.8 and y = -1.5. Completing the square is a powerful method that not only allows you to solve quadratic equations but also gives you a deeper understanding of the quadratic expressions and provides insight into the nature of their solutions.
Conclusion: Choosing the Right Method
So, guys, we've explored three powerful methods for solving the quadratic equation 10y² + 7y = 12: factoring, the quadratic formula, and completing the square. Each method has its strengths, and the best choice often depends on the specific equation and your comfort level. Factoring is usually the quickest approach when it's easily doable. The quadratic formula is a reliable choice for any quadratic equation, ensuring you always get the correct solution. Completing the square is an excellent way to deeply understand the structure of quadratic equations. By mastering these methods, you'll be well-equipped to tackle any quadratic equation that comes your way. Keep practicing and applying these techniques, and you'll become a quadratic equation wizard in no time! Keep exploring, keep learning, and don't be afraid to experiment with different approaches to find what works best for you. Happy solving!