Solve 11x² - 98x - 9 = 0: A Quadratic Equation Guide
Quadratic equations, those fascinating mathematical expressions that take the form of ax² + bx + c = 0, are fundamental to understanding many concepts in algebra and beyond. The specific equation we're diving into today is 11x² - 98x - 9 = 0. This equation, like all quadratic equations, has a unique structure that allows us to find its roots, or the values of 'x' that make the equation true. Many students encounter these equations in their academic journey, and mastering how to solve them opens doors to more complex problem-solving. Whether you're grappling with physics problems involving projectile motion, optimizing areas in geometry, or even delving into financial modeling, the ability to solve quadratic equations is an indispensable skill. We'll break down the methods to tackle 11x² - 98x - 9 = 0, making sure you feel confident and capable. Our goal is to demystify the process, moving from the general principles of quadratic equations to the specific application for this particular problem. We'll explore the most common and effective methods, ensuring that by the end of this guide, you'll not only have the solution but also a deeper understanding of why it works. This journey into solving 11x² - 98x - 9 = 0 will reinforce your algebraic prowess and equip you with a valuable tool for your mathematical arsenal. Prepare to unlock the secrets of this quadratic equation and boost your problem-solving skills!
Understanding the Quadratic Formula
Before we jump into solving our specific equation, 11x² - 98x - 9 = 0, it's crucial to have a solid grasp of the tools available. The most universal and powerful tool for solving any quadratic equation is the quadratic formula. This formula is derived from the general form ax² + bx + c = 0 and provides a direct pathway to the solutions, regardless of whether the roots are real, imaginary, or repeated. The formula itself is: x = [-b ± √(b² - 4ac)] / 2a. It looks a bit intimidating at first glance, but it's essentially a recipe for finding 'x'. Here, 'a', 'b', and 'c' are the coefficients from your quadratic equation. 'a' is the coefficient of the x² term, 'b' is the coefficient of the x term, and 'c' is the constant term. The term inside the square root, b² - 4ac, is particularly important; it's called the discriminant, and it tells us about the nature of the roots. If the discriminant is positive, there are two distinct real roots. If it's zero, there's exactly one real root (a repeated root). If it's negative, there are two complex (imaginary) roots. For our equation, 11x² - 98x - 9 = 0, we can identify a = 11, b = -98, and c = -9. Plugging these values into the quadratic formula is the most straightforward way to find the solutions for 'x'. While factoring is sometimes an option, it's not always feasible, especially with larger or more complex coefficients. The quadratic formula, however, always works. So, for 11x² - 98x - 9 = 0, the quadratic formula is our trusty companion. Let's prepare to substitute these values and perform the calculations. It's important to pay close attention to signs, especially when 'b' is negative, as squaring a negative number results in a positive number, and the double negative in '-b' will become positive. This systematic approach ensures accuracy. The beauty of the quadratic formula lies in its generality; it's a testament to the elegance of algebraic manipulation and a cornerstone of mathematical problem-solving.
Applying the Formula to 11x² - 98x - 9 = 0
Now, let's roll up our sleeves and apply the quadratic formula to our specific equation: 11x² - 98x - 9 = 0. As identified earlier, our coefficients are a = 11, b = -98, and c = -9. We'll substitute these values directly into the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a.
First, let's calculate the discriminant (b² - 4ac) to get an idea of what kind of solutions we're expecting.
Discriminant = (-98)² - 4 * (11) * (-9) Discriminant = 9604 - (-396) Discriminant = 9604 + 396 Discriminant = 10000
Since the discriminant is a positive number (10000), we know that this equation will have two distinct real roots. This is great news, as it means our solutions will be straightforward real numbers. Now, let's plug the discriminant back into the full quadratic formula:
x = [-(-98) ± √(10000)] / (2 * 11) x = [98 ± 100] / 22
This gives us two possible solutions for 'x'.
Solution 1 (using the '+' sign): x₁ = (98 + 100) / 22 x₁ = 198 / 22 x₁ = 9
Solution 2 (using the '-' sign): x₂ = (98 - 100) / 22 x₂ = -2 / 22 x₂ = -1/11
So, the solutions to the quadratic equation 11x² - 98x - 9 = 0 are x = 9 and x = -1/11. It's always a good practice to double-check these solutions by plugging them back into the original equation to ensure they hold true. For x = 9: 11(9)² - 98(9) - 9 = 11(81) - 882 - 9 = 891 - 882 - 9 = 9 - 9 = 0. For x = -1/11: 11(-1/11)² - 98(-1/11) - 9 = 11(1/121) + 98/11 - 9 = 1/11 + 98/11 - 99/11 = 99/11 - 99/11 = 0. Both solutions are correct! The application of the quadratic formula to 11x² - 98x - 9 = 0 has yielded precise results, demonstrating its power and reliability in solving quadratic equations.
Alternative Method: Factoring (When Applicable)
While the quadratic formula is our go-to method for solving 11x² - 98x - 9 = 0, it's worth mentioning that factoring is another technique that can be used to solve quadratic equations. However, factoring is only possible when the quadratic expression can be broken down into the product of two linear binomials. For 11x² - 98x - 9 = 0, we need to find two numbers that multiply to (a*c) = (11 * -9) = -99 and add up to 'b' = -98. Finding such a pair of integers can be challenging, especially with larger coefficients. In this case, the numbers are -99 and 1. If we try to factor by grouping using these numbers, we would rewrite the middle term: 11x² - 99x + 1x - 9 = 0. Then, we factor by grouping: 11x(x - 9) + 1(x - 9) = 0. This gives us (11x + 1)(x - 9) = 0. Setting each factor to zero, we get:
11x + 1 = 0 => 11x = -1 => x = -1/11 x - 9 = 0 => x = 9
As you can see, factoring also leads to the same solutions: x = 9 and x = -1/11. It's important to note that factoring requires a bit of trial and error, and it's not always straightforward or even possible for all quadratic equations, especially those with non-integer roots or irrational roots. This is why the quadratic formula is often preferred for its universality and reliability. For 11x² - 98x - 9 = 0, both methods converge to the same correct answers, offering a good opportunity to appreciate different algebraic approaches. While factoring can be quicker when it works smoothly, the quadratic formula provides a guaranteed path to the solution, making it an indispensable tool in any mathematician's toolkit. The ability to recognize when factoring is a viable option, and when to resort to the quadratic formula, is a sign of a developing mathematical intuition. Mastering both techniques empowers you to tackle a wider range of problems with confidence.
Conclusion: Mastering the Quadratic Equation
We have successfully navigated the process of solving the quadratic equation 11x² - 98x - 9 = 0 using two primary methods: the reliable quadratic formula and the sometimes quicker, though not always feasible, factoring technique. Both approaches yielded the same accurate results: x = 9 and x = -1/11. Understanding how to solve quadratic equations is a cornerstone of algebra, and it has far-reaching applications in various scientific and engineering fields. Whether you're analyzing projectile motion in physics, optimizing designs in engineering, or modeling financial markets, the ability to solve equations of the form ax² + bx + c = 0 is a critical skill.
Remember, the quadratic formula, x = [-b ± √(b² - 4ac)] / 2a, is your most powerful ally because it works for every quadratic equation. However, recognizing when an equation can be factored can sometimes save you time and effort. The key takeaway is to be comfortable with both methods and to know when to apply each one.
Practice is, as always, the key to mastery. Work through various quadratic equations, paying close attention to the coefficients and signs. Use the quadratic formula diligently, and try to spot opportunities for factoring. As you gain more experience, you'll develop an intuition for which method is most efficient for a given problem. The journey through solving 11x² - 98x - 9 = 0 is a step towards building that confidence and proficiency. Keep exploring, keep practicing, and you'll find that quadratic equations become less daunting and more like solvable puzzles.
For further exploration into the fascinating world of algebra and quadratic equations, you can visit Khan Academy, a fantastic resource for learning and practicing mathematics. You might also find valuable insights on the Wolfram MathWorld website, which offers comprehensive mathematical information.
