Hey everyone! Today, we're diving into the fascinating world of quadratic equations, specifically tackling the equation x² - 10x = 24. If you've ever felt a little intimidated by these equations, don't worry! We're going to break it down step by step, making sure you understand not just the how, but also the why behind each move. So, grab your pencils and let's get started!
Understanding Quadratic Equations
Before we jump into solving our specific equation, let's take a moment to understand what a quadratic equation actually is. In essence, a quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in our case, 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to 0. These constants determine the shape and position of the parabola when the equation is graphed. The solutions to a quadratic equation, also known as roots or zeros, are the points where the parabola intersects the x-axis. Finding these solutions is the core of what we'll be doing today.
Now, why are quadratic equations so important? Well, they pop up in all sorts of real-world scenarios! From calculating the trajectory of a ball thrown in the air to designing bridges and even predicting financial markets, quadratic equations are essential tools in mathematics, physics, engineering, and economics. Mastering them opens up a whole new world of problem-solving capabilities. The beauty of quadratic equations lies in their versatility and the multiple methods we can use to solve them. We'll explore one of the most common and effective methods: completing the square. This method not only helps us find the solutions but also provides a deeper understanding of the structure of the equation itself. So, let's keep this broader context in mind as we delve into the nitty-gritty of solving our equation. Remember, we're not just memorizing steps; we're building a solid foundation for future mathematical adventures!
Setting Up the Equation
Okay, let's get back to our specific problem: x² - 10x = 24. The first crucial step in solving any quadratic equation is to set it equal to zero. This is because most solution methods, including completing the square, rely on having the equation in the standard form (ax² + bx + c = 0). To achieve this, we need to move the constant term (24 in this case) from the right side of the equation to the left side. We do this by subtracting 24 from both sides of the equation. This maintains the equality and brings us closer to the standard form we need. So, when we subtract 24 from both sides of x² - 10x = 24, we get x² - 10x - 24 = 0. Now, our equation is in the perfect format for the next step: completing the square. This seemingly simple maneuver is a powerful technique that transforms the equation into a form that's much easier to solve. By setting the equation to zero, we've essentially created a landscape where the solutions (the roots) are the points where the quadratic function intersects the x-axis. This visual understanding can be incredibly helpful as we proceed. We're not just manipulating symbols; we're shaping the equation into a form that reveals its hidden solutions. This initial step is more than just a technicality; it's a crucial preparation for the journey ahead. By ensuring the equation is in the standard form, we set the stage for a clear and systematic solution process. So, with our equation now neatly arranged as x² - 10x - 24 = 0, we're ready to embark on the exciting adventure of completing the square!
Completing the Square: Step-by-Step
Now comes the exciting part: completing the square! This technique might sound a bit mysterious, but it's actually a clever way of rewriting a quadratic expression as a perfect square trinomial, plus or minus a constant. A perfect square trinomial is a trinomial that can be factored into the form (x + a)² or (x - a)². This transformation makes it much easier to solve for 'x'. Here's how we do it for our equation, x² - 10x - 24 = 0:
- Focus on the x² and x terms: We'll work with the x² - 10x part first. Think of this as the beginning of a perfect square.
- Find the value to complete the square: Take half of the coefficient of the 'x' term (which is -10), square it. Half of -10 is -5, and (-5)² is 25. This is the magic number we need!
- Add and subtract the value: We add and subtract 25 within the equation to maintain balance: x² - 10x + 25 - 25 - 24 = 0. Notice that we're not changing the equation's value; we're just changing its form.
- Rewrite as a perfect square: The first three terms, x² - 10x + 25, now form a perfect square trinomial. It can be factored as (x - 5)².
- Simplify the equation: Combine the constant terms: (x - 5)² - 49 = 0. Now, our equation looks much simpler and closer to a solution!
Completing the square is like taking a puzzle and rearranging its pieces to reveal a hidden picture. In this case, the picture is the solution to the equation. By carefully adding and subtracting the right value, we transform the quadratic expression into a form that allows us to isolate 'x' and find its value. This method is not just a trick; it's a powerful technique that reveals the underlying structure of the quadratic equation. It's also a versatile method that can be used to solve any quadratic equation, even those that don't factor easily. So, as we continue our journey, remember that completing the square is a valuable tool in your mathematical arsenal. It's a technique that not only helps us solve equations but also deepens our understanding of quadratic expressions and their properties. The transformation we've performed is a testament to the beauty and power of algebraic manipulation.
Isolating x and Finding the Solutions
With our equation now in the form (x - 5)² - 49 = 0, we're in the home stretch! The next step is to isolate the squared term, (x - 5)². This means we need to get it all by itself on one side of the equation. To do this, we add 49 to both sides of the equation. This gives us (x - 5)² = 49. Now, we're getting closer to unlocking the value of 'x'. The squared term is isolated, and we can see that it's equal to a perfect square. This is a crucial point because it allows us to use the square root property. The square root property states that if a² = b, then a = √b or a = -√b. In other words, we can take the square root of both sides of the equation, but we must remember to consider both the positive and negative roots. Applying this to our equation, we take the square root of both sides: √((x - 5)²) = ±√49. This simplifies to x - 5 = ±7. The ± symbol is essential because it reminds us that there are two possible solutions: one where x - 5 equals positive 7, and another where x - 5 equals negative 7. Now, we have two simple linear equations to solve:
- x - 5 = 7
- x - 5 = -7
Solving the first equation, we add 5 to both sides: x = 7 + 5, which gives us x = 12. Solving the second equation, we add 5 to both sides: x = -7 + 5, which gives us x = -2. And there we have it! We've found the two solutions to our quadratic equation: x = 12 and x = -2. These are the values of 'x' that make the equation x² - 10x = 24 true. They are also the points where the parabola represented by the equation intersects the x-axis. The journey of isolating 'x' and finding these solutions has been a testament to the power of algebraic manipulation and the importance of remembering key mathematical properties. We've navigated the steps with care, ensuring that each move brings us closer to the final answer. And now, with the solutions in hand, we can confidently say that we've conquered this quadratic equation!
Verifying the Solutions
Before we celebrate our victory, it's always a good idea to verify our solutions. This means plugging the values we found for 'x' back into the original equation to make sure they actually work. This step is crucial because it helps us catch any potential errors we might have made along the way. It's like double-checking your work on a test – it ensures accuracy and gives you peace of mind. So, let's take our solutions, x = 12 and x = -2, and plug them into the original equation: x² - 10x = 24.
First, let's check x = 12: Substituting 12 for 'x', we get (12)² - 10(12) = 144 - 120 = 24. This is exactly what we wanted! So, x = 12 is indeed a solution. Now, let's check x = -2: Substituting -2 for 'x', we get (-2)² - 10(-2) = 4 + 20 = 24. Again, this matches the right side of the equation! So, x = -2 is also a valid solution. The process of verification is more than just a formality; it's a testament to the integrity of our mathematical journey. It confirms that the steps we took were correct and that the solutions we found are accurate. By plugging the solutions back into the original equation, we're essentially putting our answers to the test. We're asking the equation, "Do these values really work?" And in both cases, the equation has answered with a resounding "Yes!" This verification step solidifies our understanding of the problem and the solution process. It reinforces the connection between the equation and its solutions, and it builds our confidence in our mathematical abilities. So, with both solutions successfully verified, we can confidently conclude that we have solved the quadratic equation x² - 10x = 24 completely and accurately.
Conclusion: The Solutions and Their Significance
So, we've successfully navigated the world of quadratic equations and found the solutions to x² - 10x = 24. The solutions, as we've shown, are x = 12 and x = -2. But what do these solutions actually mean? They are the values of 'x' that satisfy the equation, making the left side equal to the right side. Graphically, they represent the points where the parabola described by the equation intersects the x-axis. These points are also known as the roots or zeros of the quadratic function. Understanding the significance of these solutions is just as important as finding them. They provide valuable information about the behavior of the quadratic function and its relationship to the coordinate plane. The solutions tell us where the function crosses the x-axis, which can be crucial in many applications. For example, in physics, these solutions might represent the time at which a projectile hits the ground. In engineering, they could indicate the points where a structure reaches a certain height. In economics, they might represent the break-even points for a business. The ability to solve quadratic equations and interpret their solutions is a powerful skill that extends far beyond the classroom. It's a tool that can be applied to a wide range of real-world problems. The process we've used to solve this equation, completing the square, is a versatile technique that can be applied to any quadratic equation. It's a method that not only helps us find the solutions but also deepens our understanding of the structure of the equation. By mastering this technique, we've equipped ourselves with a valuable tool for solving a wide range of mathematical problems. The journey of solving x² - 10x = 24 has been more than just a mathematical exercise; it's been a journey of discovery and empowerment. We've learned how to manipulate algebraic expressions, how to apply key mathematical properties, and how to interpret the meaning of our solutions. And with this knowledge, we're ready to tackle even more challenging mathematical adventures in the future.
In conclusion, remember guys, the solution to the equation x² - 10x = 24 are 12 and -2. Keep practicing and you'll become a pro at solving quadratic equations in no time!