Solving The Quadratic Equation 5x² - 4x = 6 A Step-by-Step Guide
Hey guys! Today, we're diving deep into the fascinating world of quadratic equations, specifically tackling the equation 5x² - 4x = 6. This equation might look a bit intimidating at first glance, but don't worry, we're going to break it down step-by-step and unveil the secrets to finding its solutions. We'll not only solve it but also explore the underlying concepts and techniques involved. So, buckle up and get ready for a mathematical adventure!
The Quadratic Equation: A Quick Recap
Before we jump into solving our specific equation, let's take a moment to refresh our understanding of quadratic equations in general. A quadratic equation is a polynomial equation of the second degree. This means it involves a variable (usually 'x') raised to the power of 2, as well as possibly terms with 'x' to the power of 1 and a constant term. The general form of a quadratic equation is:
ax² + bx + c = 0
Where 'a', 'b', and 'c' are constants, and 'a' cannot be zero (otherwise, it wouldn't be a quadratic equation!). These constants play a crucial role in determining the nature and values of the solutions (also called roots) of the equation. Understanding the coefficients 'a', 'b', and 'c' is the first step towards conquering any quadratic equation that comes your way. For instance, the sign of 'a' tells us whether the parabola opens upwards (if 'a' is positive) or downwards (if 'a' is negative). The values of 'b' and 'c', in relation to 'a', influence the position and shape of the parabola, which in turn affects the roots of the equation. Remember, the roots are the points where the parabola intersects the x-axis, representing the values of 'x' that make the equation true. So, a solid grasp of these coefficients is your key to unlocking the secrets hidden within the quadratic equation.
Transforming Our Equation: Standard Form is Key
Now that we've got the basics covered, let's get back to our equation: 5x² - 4x = 6. Notice that it's not quite in the standard form we just discussed (ax² + bx + c = 0). To solve it effectively, our first mission is to rearrange it into the standard form. This involves moving all the terms to one side of the equation, leaving zero on the other side. It's like tidying up our workspace before we start a project – it makes everything much clearer and easier to handle. In our case, this means subtracting 6 from both sides of the equation. This simple move is crucial because many of our powerful tools for solving quadratic equations, like the quadratic formula, are designed to work specifically with equations in standard form. By transforming the equation, we're setting the stage for applying these techniques and finding the solutions. Think of it as translating the equation into a language that our mathematical tools can understand. Once it's in standard form, we can clearly identify the coefficients 'a', 'b', and 'c', which are the building blocks for solving the equation. So, let's make that crucial transformation and get our equation ready for action!
Performing this operation, we get:
5x² - 4x - 6 = 0
Now, we have our equation in the standard form! This is a crucial step because it allows us to easily identify the coefficients a, b, and c, which we'll need for the next step.
In this case:
- a = 5
- b = -4
- c = -6
With these values in hand, we're ready to unleash one of the most powerful tools in the quadratic equation solver's arsenal: the quadratic formula!
The Quadratic Formula: Our Hero in Shining Armor
The quadratic formula is a universal solution for any quadratic equation in the standard form. It's like a Swiss Army knife for math problems, providing a reliable method to find the roots, no matter how complex the equation might seem. This formula is derived by completing the square on the standard form of the quadratic equation, and it elegantly expresses the solutions in terms of the coefficients a, b, and c. Memorizing this formula is a must for anyone serious about mastering quadratic equations. It's not just a formula; it's a key that unlocks the solutions to a vast array of problems. When you encounter a quadratic equation that doesn't easily factor, or when you need a precise answer, the quadratic formula is your go-to tool. It eliminates the guesswork and provides a systematic way to find the roots. Think of it as a mathematical GPS, guiding you directly to the solutions, regardless of the twists and turns of the equation. So, let's bring out this hero and see how it helps us conquer our equation!
The quadratic formula states that for an equation ax² + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b² - 4ac)) / 2a
This might look a bit daunting, but don't worry! It's just a matter of plugging in the values we identified earlier.
Plugging in the Values: Let's Get to Work
Now comes the exciting part – putting our knowledge into action! We've got the quadratic formula, we've identified our coefficients (a = 5, b = -4, c = -6), so let's plug these values into the formula and watch the magic happen. This step is where the abstract concepts transform into concrete numbers, leading us closer to the solutions. It's like following a recipe – we have the ingredients (the coefficients) and the instructions (the quadratic formula), and now we're going to combine them to create the final product (the roots of the equation). Pay close attention to the signs and the order of operations, as a small mistake here can lead to a wrong answer. But don't be intimidated! With careful substitution and a bit of arithmetic, we'll unravel the solutions hidden within the formula. This is where the power of mathematics truly shines – taking a seemingly complex problem and breaking it down into manageable steps. So, let's roll up our sleeves and get those numbers plugged in!
Substituting our values into the formula, we get:
x = (-(-4) ± √((-4)² - 4 * 5 * -6)) / (2 * 5)
Simplifying the Expression: Crunching the Numbers
Okay, we've got the values plugged in, and now it's time for some good old-fashioned simplification. This is where we put our arithmetic skills to the test, carefully working through the expression to arrive at the simplest possible form. Think of it as cleaning up the equation, removing the clutter and revealing the core solution. We'll start by simplifying the expression under the square root, then deal with the rest of the terms. Remember the order of operations (PEMDAS/BODMAS) – parentheses/brackets first, then exponents/orders, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). A methodical approach is key here, as each step builds upon the previous one. By simplifying, we're not just making the expression easier to look at; we're also reducing the chances of making errors in the final calculation. So, let's take a deep breath, focus on the details, and crunch those numbers!
Let's simplify this step-by-step:
x = (4 ± √(16 + 120)) / 10
x = (4 ± √136) / 10
Simplifying the Radical: Unveiling the Hidden Simplicity
We've reached a point where we have a square root in our expression, and it's our job to simplify it as much as possible. Simplifying radicals is like finding the hidden beauty within a number – breaking it down into its prime factors and extracting any perfect squares. This not only makes the expression look cleaner but also helps us understand the nature of the solutions. To simplify √136, we need to find the largest perfect square that divides 136. This involves a bit of number sense and possibly some trial and error. Once we've found the perfect square factor, we can rewrite the radical as a product of two radicals, one of which can be simplified to a whole number. This process allows us to express the solution in its most elegant form, revealing the underlying mathematical structure. So, let's put on our detective hats and uncover the hidden simplicity within √136!
Now, we need to simplify the radical √136. We can factor 136 as 4 * 34, where 4 is a perfect square.
So, √136 = √(4 * 34) = √4 * √34 = 2√34
Substituting this back into our equation, we get:
x = (4 ± 2√34) / 10
Final Simplification: Putting the Pieces Together
We're almost there! We've simplified the radical, and now it's time for the final touches – simplifying the entire expression. This involves looking for common factors in the numerator and denominator and canceling them out. It's like putting the final brushstrokes on a painting, refining the details to create a polished and elegant result. In our case, we can see that both the numerator and the denominator have a common factor of 2. Dividing both by 2 will give us the simplest possible form of the solution. This final simplification is crucial not just for aesthetics but also for clarity. A simplified answer is easier to understand and work with in further calculations. It's the culmination of all our efforts, bringing us to the final answer. So, let's put the pieces together and complete our mathematical journey!
We can see that all terms have a common factor of 2. Dividing both the numerator and denominator by 2, we get:
x = (2 ± √34) / 5
The Solutions: Our Final Answer
And there you have it, guys! We've successfully navigated the quadratic equation, battled the square roots, and emerged victorious with our solutions. The solutions to the equation 5x² - 4x = 6 are:
x = (2 + √34) / 5 and x = (2 - √34) / 5
These are the two values of 'x' that make the equation true. They represent the points where the parabola defined by the equation intersects the x-axis. This journey through the quadratic equation highlights the power of mathematical tools and techniques. We've seen how the quadratic formula can be used to solve any quadratic equation, and we've practiced the important skills of simplifying expressions and radicals. But more than just finding the answers, we've gained a deeper understanding of the underlying concepts and the beauty of mathematical problem-solving. So, celebrate your success, and get ready to tackle the next mathematical challenge!
Alternative form of the solution
The solution can also be written as
x = 2/5 ± (√34)/ 5
Key Takeaways: What We Learned Today
Wow, what a journey! We've successfully conquered the quadratic equation 5x² - 4x = 6, and along the way, we've learned some valuable lessons. Let's recap the key takeaways from our mathematical adventure. First and foremost, we revisited the quadratic formula, a powerful tool for solving any equation in the form ax² + bx + c = 0. We saw how this formula provides a systematic way to find the solutions, regardless of the complexity of the equation. We also emphasized the importance of transforming the equation into standard form before applying the formula. This step is crucial for correctly identifying the coefficients a, b, and c, which are the building blocks for the quadratic formula. Furthermore, we honed our skills in simplifying expressions, including radicals. Simplifying radicals not only makes the answer look cleaner but also reveals the underlying mathematical structure. Finally, we learned the importance of careful calculation and attention to detail. A small mistake in arithmetic can lead to a wrong answer, so it's crucial to double-check our work and proceed methodically. But beyond the specific steps and techniques, we also gained a deeper appreciation for the beauty and power of mathematics. We saw how abstract concepts can be applied to solve concrete problems, and we experienced the satisfaction of unraveling a complex equation to arrive at a clear and elegant solution. So, remember these lessons, and keep exploring the fascinating world of mathematics!
Practice Makes Perfect: Your Turn to Shine
Now that we've cracked the code for solving 5x² - 4x = 6, it's your turn to put your skills to the test! The best way to master quadratic equations is through practice, so let's dive into some exercises that will help you solidify your understanding. I encourage you, guys, to try solving different quadratic equations using the techniques we've discussed today. Start with simpler equations to build your confidence, and then gradually tackle more challenging ones. Remember, the key is to follow a systematic approach: transform the equation into standard form, identify the coefficients, apply the quadratic formula, simplify the expression, and double-check your work. Don't be afraid to make mistakes – they're a natural part of the learning process. Each mistake is an opportunity to learn and grow. And if you get stuck, don't hesitate to review the steps we've covered or seek help from a teacher, tutor, or online resources. The more you practice, the more comfortable and confident you'll become in solving quadratic equations. So, grab a pencil and paper, and let's get those mathematical muscles working! Remember, every problem you solve brings you one step closer to mastering this essential mathematical skill.
Here are a few equations you can try:
- 2x² + 5x - 3 = 0
- x² - 4x + 4 = 0
- 3x² - 2x - 1 = 0
Conclusion: The Power of Mathematical Problem-Solving
We've reached the end of our journey through the equation 5x² - 4x = 6, and what a rewarding journey it has been! We started with a seemingly complex problem, but by breaking it down into manageable steps, we were able to find the solutions and gain a deeper understanding of quadratic equations. We've not only learned how to solve a specific equation but also developed valuable problem-solving skills that can be applied to a wide range of mathematical challenges. We've seen the power of the quadratic formula, the importance of simplifying expressions, and the value of perseverance in the face of difficulty. But perhaps the most important takeaway is the realization that mathematics is not just about numbers and formulas; it's about logical thinking, creative problem-solving, and the joy of discovery. Every equation solved, every concept mastered, is a step forward on a path of intellectual growth and empowerment. So, embrace the challenges, celebrate the successes, and continue to explore the fascinating world of mathematics! Keep practicing, keep learning, and never stop questioning. The power of mathematical problem-solving is within you!
