Solve Quadratic Equations: $x^2 - 6 = 2 - 18x$

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Solving quadratic equations is a fundamental skill in mathematics, opening doors to understanding countless real-world phenomena. From physics to finance, these equations provide powerful tools for modeling and prediction. If you've ever faced an equation like $x^2 - 6 = 2 - 18x$ and wondered how to find its solutions, you're in the right place! This comprehensive guide will walk you through the entire process, demystifying each step and ensuring you gain a solid grasp of finding solutions to such algebraic challenges. We'll dive deep into transforming the equation, applying the trusty quadratic formula, and simplifying the result to uncover the exact values of 'x' that satisfy the expression. Get ready to boost your algebraic prowess and conquer quadratic equations with confidence. Our goal is to not only arrive at the correct answer, $x = -9 \pm \sqrt{89}$, but to also equip you with the understanding needed to tackle similar problems effortlessly, making solving quadratic equations an intuitive and straightforward task rather than a daunting one. We believe in building a strong foundation, which means we won't just give you the answer; we'll show you the 'why' behind every mathematical move, illustrating how each manipulation brings us closer to the desired solutions. By the end of this article, you'll be well-versed in the techniques required to solve equations of this form, and you'll appreciate the elegance and utility of the methods involved. Let's embark on this mathematical journey together and unlock the secrets held within this quadratic expression.

Understanding Quadratic Equations: A Foundation

Before we can effectively begin solving quadratic equations, it's absolutely essential to grasp what they are and why they're so crucial in mathematics and beyond. A quadratic equation is any equation that can be rearranged into the standard form: ax² + bx + c = 0, where 'x' represents an unknown variable, and 'a', 'b', and 'c' are coefficients, with 'a' never being equal to zero. If 'a' were zero, it would simply be a linear equation, not a quadratic one. These equations are characterized by the highest power of the variable being two (hence, 'quad' from the Latin for square). Their significance stretches far and wide, making them indispensable in various scientific and engineering fields. Think about the trajectory of a thrown ball—that's modeled by a quadratic equation. The design of parabolic antennas, the optimization of production costs, or even understanding the growth patterns in certain biological systems often involve quadratic equations. Knowing how to find solutions to these equations means you can predict outcomes, design efficient systems, and solve complex problems across disciplines. While there are several methods for solving quadratic equations, such as factoring, completing the square, and graphical methods, the quadratic formula stands out as the most universal and reliable approach. It's a true workhorse, guaranteeing a solution for any quadratic equation, regardless of how complex the coefficients might be. This powerful formula provides a direct path to the values of 'x', even when factoring seems impossible or completing the square becomes too cumbersome. For our specific problem, $x^2 - 6 = 2 - 18x$, we'll rely on this robust formula to guide us to the correct answers, showing you its incredible power in action. Mastering this foundational understanding will empower you to tackle not just this problem, but a vast array of mathematical challenges that depend on quadratic principles. It's about building a versatile toolkit for problem-solving, and the quadratic formula is a prime piece of that kit. The ability to correctly identify the components (a, b, c) and plug them into the formula is a skill that will serve you well throughout your mathematical journey.

Step-by-Step Solution: Unraveling $x^2 - 6 = 2 - 18x$

Now that we have a solid understanding of what quadratic equations are, let's roll up our sleeves and dive into the practical process of solving quadratic equations with our specific example: $x^2 - 6 = 2 - 18x$. This equation might look a little intimidating at first, but by following a structured, step-by-step approach, we can break it down into manageable pieces and methodically work towards finding solutions. The key here is organization and careful attention to algebraic rules. We'll transform this equation into the standard quadratic form, identify its coefficients, and then unleash the power of the quadratic formula to derive the exact values of 'x'. Each step is crucial, building upon the last to ensure accuracy and clarity in our solution. This methodical approach is what makes complex algebraic problems tractable. By the end of this section, you'll not only have the answer but also a deeper appreciation for the logical progression involved in advanced algebra, specifically when working with solving quadratic equations. We're aiming for more than just a quick fix; we're building a sustainable problem-solving skill set.

Step 1: Standardizing the Equation

Our very first move in solving quadratic equations is to get our given equation, $x^2 - 6 = 2 - 18x$, into the familiar standard form: ax² + bx + c = 0. This standardization is absolutely critical because the quadratic formula is designed specifically to work with this format. Think of it as preparing your ingredients before you start cooking; you need everything in its proper place. Currently, the terms are scattered across both sides of the equals sign, making it difficult to directly apply any formula. Our goal is to gather all the terms on one side, typically the left side, and set the entire expression equal to zero. This involves using basic algebraic operations: addition and subtraction. Let's start by moving the terms from the right side of the equation to the left. We have '2' and '-18x' on the right. To move '2' to the left, we subtract '2' from both sides of the equation. Simultaneously, to move '-18x' to the left, we add '18x' to both sides. Performing these operations, the equation transforms as follows:

$x^2 - 6 = 2 - 18x$

First, add $18x$ to both sides:

$x^2 - 6 + 18x = 2 - 18x + 18x$

This simplifies to:

$x^2 + 18x - 6 = 2$

Next, subtract $2$ from both sides:

$x^2 + 18x - 6 - 2 = 2 - 2$

Which gives us:

$x^2 + 18x - 8 = 0$

Voilà! We now have our equation in the perfect standard form: ax² + bx + c = 0. From this standardized form, we can clearly identify the coefficients 'a', 'b', and 'c', which are the numerical values that precede our variables and the constant term. In our transformed equation, $x^2 + 18x - 8 = 0$, we can see that:

• The coefficient of $x^2$ is a = 1 (since $x^2$ is the same as $1x^2$).
• The coefficient of $x$ is b = 18.
• The constant term is c = -8.

Accurately identifying these values is a crucial prerequisite for the next step, where we will insert them into the quadratic formula. Any mistake here will lead to incorrect solutions. Take your time with this initial rearrangement; it lays the groundwork for the rest of our problem-solving journey in finding solutions to this quadratic equation. This step highlights the importance of meticulous algebra in solving quadratic equations and ensures we're setting up the problem correctly for the powerful formula ahead. Without this proper setup, even the most robust formula cannot yield the correct results.

Step 2: Applying the Quadratic Formula

With our equation neatly in standard form ($x^2 + 18x - 8 = 0$) and our coefficients identified ($a=1$, $b=18$, $c=-8$), we're now ready for the star of the show: the quadratic formula. This formula is a true hero for solving quadratic equations, as it provides a direct, foolproof method for finding solutions regardless of the complexity of the numbers involved. The formula itself is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. It might look a bit daunting at first glance, but it's essentially a plug-and-play tool. All we need to do is carefully substitute our values for 'a', 'b', and 'c' into their respective places within the formula. Let's go through this substitution process step-by-step to ensure precision. We'll replace 'b' with 18, 'a' with 1, and 'c' with -8:

$x = \frac{-(18) \pm \sqrt{(18)^2 - 4(1)(-8)}}{2(1)}$

Now, let's start simplifying the expression, beginning with the terms inside the square root, which is often referred to as the discriminant ($b^2 - 4ac$). The discriminant is incredibly important because it tells us about the nature of the solutions (real, imaginary, or repeated).

Calculate $b^2$: $(18)^2 = 18 \times 18 = 324$

Calculate $4ac$: $4(1)(-8) = -32$

Now, substitute these back into the discriminant part of the formula:

$b^2 - 4ac = 324 - (-32) = 324 + 32 = 356$

Notice the double negative here: subtracting a negative number is equivalent to adding a positive one. This is a common place for small errors, so always be extra careful with your signs! The value of our discriminant is 356. Since 356 is a positive number, we know we will have two distinct real solutions for 'x', which is a good sign for finding solutions to our quadratic equation. Now, let's plug this back into the larger quadratic formula:

$x = \frac{-18 \pm \sqrt{356}}{2}$

We are now one step closer to our final answer. The application of the formula is complete, and the next phase involves simplifying the square root and the entire fraction. This step really highlights the power of the quadratic formula in systematically solving quadratic equations, transforming a complex expression into a manageable numerical problem. Understanding how each part of the formula contributes to the solution is key to truly mastering this fundamental algebraic tool.

Step 3: Simplifying the Solution

We've arrived at the point where our solution for 'x' looks like this: $x = \frac{-18 \pm \sqrt{356}}{2}$. The final hurdle in solving quadratic equations is to simplify this expression as much as possible to present our finding solutions in their most elegant and understandable form. This often involves simplifying the square root and then reducing the entire fraction if common factors exist. First, let's focus on $\sqrt{356}$. To simplify a square root, we look for perfect square factors within the number. We can start by testing small prime numbers or even powers of two.

Is 356 divisible by 4 (a perfect square)? $356 \div 4 = 89$

So, we can rewrite $\sqrt{356}$ as $\sqrt{4 \times 89}$.

Using the property of square roots that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, we get:

$\sqrt{4 \times 89} = \sqrt{4} \times \sqrt{89} = 2\sqrt{89}$

Now, we need to determine if 89 can be simplified further. To do this, we check if 89 has any perfect square factors other than 1. We can test prime numbers: 2, 3, 5, 7, 11...

• 89 is not divisible by 2 (it's odd).
• The sum of its digits (8+9=17) is not divisible by 3, so 89 is not divisible by 3.
• It doesn't end in 0 or 5, so it's not divisible by 5.
• $89 \div 7 = 12$ with a remainder, so not divisible by 7.
• For 11, $11^2 = 121$, which is greater than 89, so we only need to check primes up to $\sqrt{89}$, which is approximately 9.4. Since we've checked all primes up to 7, and 89 is not divisible by any of them, 89 is a prime number. Therefore, $\sqrt{89}$ cannot be simplified further.

So, we replace $\sqrt{356}$ with $2\sqrt{89}$ in our solution:

$x = \frac{-18 \pm 2\sqrt{89}}{2}$

Now, we can see that both terms in the numerator ($-18$ and $2\sqrt{89}$) are divisible by 2, and the denominator is also 2. This means we can simplify the entire fraction by dividing each term in the numerator by the denominator. It's crucial to divide both terms by 2, not just one!

$x = \frac{2(-9 \pm \sqrt{89})}{2}$

Now, cancel out the common factor of 2 in the numerator and denominator:

$x = -9 \pm \sqrt{89}$

And there we have it! The solutions to the equation $x^2 - 6 = 2 - 18x$ are $x = -9 + \sqrt{89}$ and $x = -9 - \sqrt{89}$. These are two distinct real, irrational solutions. Comparing this result to the given options, we find that our solution perfectly matches option D. This final simplification step is essential for presenting clear, concise, and correct answers when finding solutions for any quadratic equation, solidifying our understanding of the process and demonstrating mastery in solving quadratic equations from start to finish. This detailed approach ensures that no stone is left unturned in our quest for mathematical clarity and accuracy.

Why This Method Works: The Power of the Quadratic Formula

When it comes to solving quadratic equations, the quadratic formula isn't just a recipe; it's a profound mathematical truth derived from the process of completing the square. Its sheer power lies in its universality: it works for any quadratic equation in the form $ax^2 + bx + c = 0$, even those that seem impossible to factor or too messy for other methods. This reliability is what makes it a cornerstone of algebra and a tool that every student should master. The formula provides a direct path to finding solutions, eliminating the guesswork often associated with factoring, and streamlining the process when dealing with complicated coefficients or irrational roots, as we saw with $\sqrt{89}$ in our example. Imagine trying to factor $x^2 + 18x - 8 = 0$; you'd be looking for two numbers that multiply to -8 and add to 18. Such integer pairs don't exist, which means factoring wouldn't yield clean, straightforward integer solutions. This is precisely where the quadratic formula shines, providing the exact irrational solutions with elegance and efficiency. The term under the square root, $b^2 - 4ac$, the discriminant, offers invaluable insight into the nature of the solutions. If it's positive, as it was in our case (356), you get two distinct real solutions. If it's zero, you get exactly one real (repeated) solution. If it's negative, you get two complex (imaginary) solutions. This predictive power is another testament to the formula's brilliance, guiding our expectations before we even complete the calculation. While methods like factoring are excellent for certain types of quadratics, especially those with easily identifiable integer roots, they fall short when the roots are irrational or complex. Completing the square is the method from which the formula is derived, and while effective, it can be more laborious for general use. Therefore, for reliably solving complex quadratic equations and consistently finding solutions in all scenarios, the quadratic formula remains the gold standard. Its systematic approach ensures that you always arrive at the correct answer, reinforcing its status as an indispensable tool in your mathematical arsenal. It transforms what could be a frustrating trial-and-error process into a logical, step-by-step derivation of the solution, empowering you to confidently tackle any quadratic challenge that comes your way. Understanding this underlying strength makes the formula not just a memorized sequence, but a deeply appreciated instrument for mathematical exploration.

Beyond the Classroom: Real-World Applications of Quadratic Equations

While solving quadratic equations might seem like a purely academic exercise, confined to textbooks and classrooms, their applications extend far beyond, permeating various aspects of our daily lives and technological advancements. Quadratic equations are not just abstract mathematical constructs; they are powerful tools for modeling and understanding the world around us. In the realm of physics and engineering, quadratic equations are absolutely indispensable. Consider the trajectory of a projectile, whether it's a football kicked across a field, a cannonball fired from a historic fortress, or even a rocket launching into space. The path these objects follow is a parabola, which can be precisely described by a quadratic equation. Engineers use these equations to design bridges, calculate stress points in materials, and optimize the shape of structures for maximum stability and efficiency. For instance, the arch of a bridge or the curve of a satellite dish are often parabolic in shape, and their design relies heavily on finding solutions to quadratic equations to ensure optimal performance and safety. In sports, understanding quadratic motion can help athletes and coaches analyze throws, kicks, and jumps to improve performance. The flight of a golf ball, the arc of a basketball shot, or the path of a long jump can all be modeled quadratically, allowing for precise calculations of distance, height, and time. Even in finance and economics, quadratic models are used to forecast market trends, calculate profits and losses, and optimize investment portfolios. For example, businesses might use quadratic equations to determine the optimal price point for a product to maximize revenue, considering how demand changes with price. If the relationship between price and quantity sold follows a parabolic curve, then solving quadratic equations becomes key to identifying the peak profit. Furthermore, in computer graphics and animation, quadratic equations are fundamental for rendering curves, creating realistic motions, and designing intricate visual effects. From designing roller coasters to simulating the spread of a disease, the ability to solve quadratic equations allows us to predict outcomes, design efficient systems, and solve complex, real-world problems. They enable us to turn abstract numerical relationships into tangible, impactful solutions, making them far more than just mathematical puzzles; they are essential instruments for innovation and understanding. These practical examples underscore why mastering finding solutions to quadratic equations is a valuable skill, equipping individuals with the analytical prowess needed to tackle challenges across a multitude of disciplines, illustrating that algebra is truly a language for describing the physical and economic world.

Conclusion

Congratulations! You've successfully navigated the journey of solving quadratic equations with our example, $x^2 - 6 = 2 - 18x$. We started by transforming the equation into its standard form, $x^2 + 18x - 8 = 0$, carefully identifying the coefficients $a=1$, $b=18$, and $c=-8$. Then, we masterfully applied the powerful quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, to systematically arrive at the intermediate solution $x = \frac{-18 \pm \sqrt{356}}{2}$. Finally, through meticulous simplification of the square root and the entire fraction, we found the elegant and precise solutions: $x = -9 \pm \sqrt{89}$. This process of finding solutions not only provided the answer to our specific problem but also reinforced the fundamental steps and reasoning behind solving any quadratic equation. The ability to methodically rearrange, apply a formula, and simplify results is a cornerstone of algebraic proficiency, and it's a skill that will empower you in countless mathematical and real-world scenarios. Remember, practice is key to mastery! The more you engage with solving quadratic equations, the more intuitive and natural the process will become. Keep exploring, keep questioning, and keep building your mathematical confidence. This fundamental mathematical skill will undoubtedly serve you well in your academic and professional endeavors, allowing you to confidently tackle future challenges where quadratic relationships are at play. Embrace the logic, and the solutions will follow.

For further learning and to deepen your understanding of quadratic equations and other algebraic concepts, we highly recommend exploring these trusted resources:

• Khan Academy's Algebra II Quadratic Equations
• Wolfram MathWorld on Quadratic Equation
• Math Is Fun: Quadratic Equations