Solving $5x^2 + 7x - 6 = 0$ A Comprehensive Guide

Hey guys! Today, we're going to tackle a classic quadratic equation: $5x^2 + 7x - 6 = 0$. Quadratic equations pop up everywhere in math and real-world applications, so mastering how to solve them is super important. We'll explore different methods to crack this equation and understand the underlying principles. So, grab your pencils, and let's dive in!

Understanding Quadratic Equations

Before we jump into solving, let's make sure we're all on the same page about what a quadratic equation actually is. In its most general form, a quadratic equation looks like this: $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are constants, and 'x' is our variable. The key characteristic of a quadratic equation is the $x^2$ term, which gives it that U-shaped curve when graphed (a parabola). The solutions to a quadratic equation, also known as roots or zeros, are the x-values where the parabola intersects the x-axis. These solutions tell us where the equation equals zero. For our specific equation, $5x^2 + 7x - 6 = 0$, we have $a = 5$, $b = 7$, and $c = -6$. Now that we've identified these coefficients, we can start thinking about how to solve for 'x'. Solving quadratic equations is a fundamental skill in algebra, and there are several methods we can use. We'll primarily focus on two powerful techniques: factoring and the quadratic formula. Factoring involves breaking down the quadratic expression into a product of two binomials, which can then be easily solved. The quadratic formula, on the other hand, is a universal tool that works for any quadratic equation, regardless of whether it can be easily factored. Understanding these methods is crucial not only for solving equations but also for grasping broader mathematical concepts. The solutions to quadratic equations have practical applications in fields such as physics, engineering, and economics, where they can model real-world phenomena like projectile motion, optimization problems, and financial modeling.

Method 1: Factoring the Quadratic Equation

Let's kick things off with the factoring method, which is often the quickest way to solve a quadratic equation if it's factorable. The main idea behind factoring is to rewrite the quadratic expression as a product of two binomials. To factor $5x^2 + 7x - 6 = 0$, we need to find two binomials of the form $(px + q)(rx + s)$ such that when they are multiplied together, they give us our original quadratic expression. This involves a bit of trial and error, but there's a systematic approach we can follow. First, we look at the leading coefficient (5) and the constant term (-6). We need to find two numbers that multiply to give us the product of these two values, which is $5 imes -6 = -30$. At the same time, these two numbers must also add up to the middle coefficient, which is 7. After a little thought, we can identify these two numbers as 10 and -3, because $10 imes -3 = -30$ and $10 + (-3) = 7$. Now, we rewrite the middle term (7x) using these two numbers: $5x^2 + 10x - 3x - 6 = 0$. Next, we use a technique called factoring by grouping. We group the first two terms and the last two terms together: $(5x^2 + 10x) + (-3x - 6) = 0$. From the first group, we can factor out a $5x$, and from the second group, we can factor out a -3: $5x(x + 2) - 3(x + 2) = 0$. Notice that we now have a common factor of $(x + 2)$ in both terms. We can factor this out, giving us $(5x - 3)(x + 2) = 0$. Now we have our quadratic equation in factored form! To find the solutions, we simply set each factor equal to zero and solve for x: $5x - 3 = 0$ or $x + 2 = 0$. Solving these linear equations, we get x = rac{3}{5} and $x = -2$. These are the two solutions to our quadratic equation. Factoring is a powerful technique because it breaks down a complex problem into simpler parts. When you can spot the factors, it's often the fastest way to solve a quadratic equation. However, not all quadratic equations are easily factorable, which is where our next method comes in handy.

Method 2: The Quadratic Formula

When factoring seems tricky or impossible, the quadratic formula comes to the rescue! This formula is a universal tool that can solve any quadratic equation, regardless of its coefficients. It might look a bit intimidating at first, but it's actually quite straightforward to use once you get the hang of it. Remember our general form of a quadratic equation: $ax^2 + bx + c = 0$. The quadratic formula states that the solutions for 'x' are given by: x = rac{-b old{\pm} ext{sqrt}(b^2 - 4ac)}{2a}. Let's break this down and see how it applies to our equation, $5x^2 + 7x - 6 = 0$. We've already identified that $a = 5$, $b = 7$, and $c = -6$. Now, we simply plug these values into the quadratic formula: x = rac{-7 old{\pm} ext{sqrt}(7^2 - 4 imes 5 imes -6)}{2 imes 5}. First, let's simplify the expression under the square root: $7^2 - 4 imes 5 imes -6 = 49 + 120 = 169$. So, we have x = rac{-7 old{\pm} ext{sqrt}(169)}{10}. The square root of 169 is 13, so our equation becomes x = rac{-7 old{\pm} 13}{10}. Now, we have two possible solutions, one with the plus sign and one with the minus sign: x_1 = rac{-7 + 13}{10} = rac{6}{10} = rac{3}{5} and x_2 = rac{-7 - 13}{10} = rac{-20}{10} = -2. Hey, look at that! We got the same solutions as we did with factoring: x = rac{3}{5} and $x = -2$. This confirms that the quadratic formula is a reliable method for solving quadratic equations. The quadratic formula is particularly useful when dealing with equations that have irrational or complex solutions, which can be difficult or impossible to find through factoring. The term inside the square root, $b^2 - 4ac$, is called the discriminant. The discriminant tells us a lot about the nature of the solutions. If the discriminant is positive, we have two distinct real solutions (as in our case). If it's zero, we have one real solution (a repeated root). And if it's negative, we have two complex solutions. Understanding the quadratic formula and the discriminant provides a complete toolkit for analyzing and solving quadratic equations.

Verifying the Solutions

It's always a good practice to check your answers, especially in math. We found two solutions for our equation, x = rac{3}{5} and $x = -2$. To verify these solutions, we'll plug them back into the original equation, $5x^2 + 7x - 6 = 0$, and see if they make the equation true. Let's start with x = rac{3}{5}: 5(rac{3}{5})^2 + 7(rac{3}{5}) - 6 = 5(rac{9}{25}) + rac{21}{5} - 6 = rac{9}{5} + rac{21}{5} - 6 = rac{30}{5} - 6 = 6 - 6 = 0. So, x = rac{3}{5} is indeed a solution. Now, let's check $x = -2$: $5(-2)^2 + 7(-2) - 6 = 5(4) - 14 - 6 = 20 - 14 - 6 = 0$. This solution also checks out! By verifying our solutions, we can be confident that we've solved the equation correctly. This step is a crucial part of the problem-solving process, as it helps prevent errors and reinforces our understanding of the concepts involved. Guys, always remember to verify your solutions whenever possible – it's a small step that can make a big difference in your accuracy and confidence. In mathematics, verification is not just about confirming the answer; it's about ensuring the logical consistency and integrity of the entire solution process. It's a way of double-checking your work and building a deeper understanding of the underlying principles.

Conclusion

Alright, guys, we've successfully solved the quadratic equation $5x^2 + 7x - 6 = 0$ using two different methods: factoring and the quadratic formula. We found that the solutions are x = rac{3}{5} and $x = -2$. We also took the time to verify these solutions, giving us extra confidence in our results. Solving quadratic equations is a fundamental skill in algebra, and mastering these techniques will open doors to more advanced mathematical concepts and real-world applications. Whether you prefer the elegance of factoring or the reliability of the quadratic formula, having both methods in your toolkit will make you a more versatile problem-solver. Remember, practice makes perfect! The more you work with quadratic equations, the more comfortable and confident you'll become. So, don't be afraid to tackle new problems and explore different approaches. And if you ever get stuck, don't hesitate to review the steps we've covered or seek out additional resources. Math is a journey, and every problem you solve is a step forward. Keep practicing, keep learning, and you'll be amazed at what you can achieve! We have explored factoring, which involves breaking down the quadratic expression into binomial factors, and the quadratic formula, a universal tool that works for all quadratic equations. Both methods yielded the same solutions, highlighting the versatility and power of these techniques. This detailed exploration not only solves the specific equation but also provides a comprehensive understanding of how to approach and solve quadratic equations in general. Keep up the great work, and happy problem-solving!