Solve The Equation $x^2 - 12x + 36 = 90$ For $x$.

Solving quadratic equations is a fundamental skill in algebra, and the equation $x^2 - 12x + 36 = 90$ provides a great example to illustrate various methods. In this comprehensive guide, we will walk through the process of solving this equation step by step, highlighting key concepts and techniques. Our primary goal is to find the values of $x$ that satisfy the equation, and we will explore different approaches to achieve this. Understanding how to solve quadratic equations is crucial not only for academic success but also for real-world applications in fields such as physics, engineering, and finance. We will also delve into the logic behind each step, ensuring a clear understanding of the underlying principles. Furthermore, we'll discuss common mistakes to avoid and strategies for checking your solutions. By the end of this guide, you will have a solid grasp of how to solve quadratic equations and be well-equipped to tackle similar problems.

The journey to solving a quadratic equation often begins with recognizing its standard form, which is $ax^2 + bx + c = 0$. The given equation, $x^2 - 12x + 36 = 90$, isn't immediately in this form, but we can easily transform it. The first step involves moving all terms to one side of the equation, leaving zero on the other side. This is crucial because many solution methods, such as factoring and the quadratic formula, rely on this standard form. To achieve this, we subtract 90 from both sides of the equation. This maintains the balance of the equation and brings us closer to the standard form we need. The resulting equation will be a clear representation of a quadratic expression set equal to zero, allowing us to apply various techniques to find the values of $x$ that make the equation true. This transformation is a foundational step in solving any quadratic equation, and mastering it is essential for success in algebra.

After subtracting 90 from both sides, we obtain the equation $x^2 - 12x + 36 - 90 = 0$. Simplifying this, we get $x^2 - 12x - 54 = 0$. Now, our quadratic equation is in the standard form $ax^2 + bx + c = 0$, where $a = 1$, $b = -12$, and $c = -54$. This standard form is crucial for applying various methods to solve the equation, such as factoring, completing the square, or using the quadratic formula. Identifying the coefficients $a$, $b$, and $c$ is a key step, especially when using the quadratic formula, as these values are directly substituted into the formula. Understanding the role of each coefficient in the quadratic equation helps in choosing the most efficient method for solving it. For instance, if the coefficients allow for easy factoring, that might be the quickest route. If not, the quadratic formula provides a reliable alternative. Thus, transforming the equation into standard form and identifying the coefficients is a fundamental step in the problem-solving process.

Method 1: Completing the Square

Completing the square is a powerful technique for solving quadratic equations. It involves transforming the quadratic expression into a perfect square trinomial, which can then be easily solved. To apply this method to our equation, $x^2 - 12x - 54 = 0$, we first focus on the terms involving $x$, which are $x^2$ and $-12x$. We aim to rewrite these terms as part of a perfect square. A perfect square trinomial is of the form $(x + k)^2$ or $(x - k)^2$, where $k$ is a constant. Expanding these forms, we get $x^2 + 2kx + k^2$ and $x^2 - 2kx + k^2$, respectively. By comparing the coefficient of the $x$ term in our equation (-12) with the general form (±2k), we can determine the value of $k$ needed to complete the square. This process allows us to manipulate the equation in a way that makes it easier to isolate $x$ and find its values. Completing the square is not only a method for solving equations but also a valuable technique in various areas of mathematics, such as calculus and coordinate geometry.

To complete the square for $x^2 - 12x - 54 = 0$, we take half of the coefficient of the $x$ term, which is $-12$, and square it. Half of $-12$ is $-6$, and $(-6)^2 = 36$. Notice that the first two terms of our equation, $x^2 - 12x$, already resemble the beginning of the perfect square $(x - 6)^2$, which expands to $x^2 - 12x + 36$. The key insight here is that we need to add and subtract 36 to the left side of the equation to maintain its balance while creating a perfect square trinomial. This step is crucial because it allows us to rewrite the quadratic expression in a more manageable form. By adding and subtracting the same value, we are essentially adding zero, which doesn't change the equation's solution but transforms its structure. This manipulation is the heart of the completing the square method, enabling us to convert a general quadratic equation into one that can be easily solved by taking square roots.

Adding and subtracting 36 to the left side of the equation, we get $x^2 - 12x + 36 - 36 - 54 = 0$. Now we can rewrite the first three terms as a perfect square: $(x - 6)^2 - 36 - 54 = 0$. Combining the constant terms, we have $(x - 6)^2 - 90 = 0$. This form of the equation is much easier to work with because it isolates the $x$ term within a square. The next step involves isolating the squared term by adding 90 to both sides of the equation. This brings us closer to solving for $x$ by undoing the operations that are applied to it. The process of completing the square has transformed the original quadratic equation into a form where we can directly apply the square root property, which is a fundamental technique for solving equations involving squared terms. By carefully manipulating the equation, we have set the stage for a straightforward solution.

Adding 90 to both sides of the equation $(x - 6)^2 - 90 = 0$, we get $(x - 6)^2 = 90$. Now we can take the square root of both sides. Remember to consider both the positive and negative square roots, as both will satisfy the equation. Taking the square root gives us $x - 6 = \\\pm \\sqrt{90}$. This is a crucial step, as it acknowledges that both positive and negative roots can lead to valid solutions for $x$. The $\\\\pm$ symbol is a shorthand way of representing this dual possibility. It's a common mistake to forget the negative root, which would lead to an incomplete solution. The square root of 90 can be simplified, which we will do in the next step. This step of taking the square root is a direct consequence of the completing the square method, which transforms the quadratic equation into a form where this operation can be readily applied. By considering both positive and negative roots, we ensure a comprehensive solution to the equation.

We can simplify $\\\\pm \\sqrt{90}$ by factoring out the largest perfect square from 90. Since $90 = 9 imes 10$, and 9 is a perfect square, we can rewrite $\\\\sqrt{90}$ as $\\\\sqrt{9 imes 10} = \\\\sqrt{9} imes \\\\sqrt{10} = 3\\\\sqrt{10}$. Therefore, $x - 6 = \\\\pm 3\\\\sqrt{10}$. To isolate $x$, we add 6 to both sides of the equation, resulting in $x = 6 \\\\pm 3\\\\sqrt{10}$. This gives us two possible solutions for $x$: $x = 6 + 3\\\\sqrt{10}$ and $x = 6 - 3\\\\sqrt{10}$. These solutions represent the points where the quadratic function intersects the x-axis, and they are the values of $x$ that make the original equation true. The simplification of the square root is an important step in presenting the solution in its simplest form. By isolating $x$, we have successfully solved the quadratic equation using the completing the square method.

Method 2: Using the Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations of the form $ax^2 + bx + c = 0$. It provides a direct way to find the solutions for $x$, regardless of whether the equation can be easily factored or completed the square. The formula is given by: $x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}$. To apply this formula to our equation, $x^2 - 12x - 54 = 0$, we first identify the coefficients $a$, $b$, and $c$. As we determined earlier, $a = 1$, $b = -12$, and $c = -54$. The quadratic formula is a powerful tool because it guarantees a solution for any quadratic equation, even those with complex roots. Understanding and correctly applying the quadratic formula is a fundamental skill in algebra. It's also essential to remember the formula itself, as it is a cornerstone in solving quadratic equations. The formula essentially encapsulates the process of completing the square, providing a shortcut to the solutions.

Substituting the values of $a$, $b$, and $c$ into the quadratic formula, we get: $x = \\frac-(-12) \\pm \\sqrt{(-12)^2 - 4(1)(-54)}}{2(1)}$. This step involves careful substitution to ensure that each value is placed correctly in the formula. The negative signs, in particular, need close attention to avoid errors. After substitution, the equation becomes $x = \\frac{12 \\pm \\sqrt{144 + 216}{2}$. The next step is to simplify the expression under the square root and the rest of the equation. Correct substitution is crucial because it sets the stage for the rest of the calculation. An error in this step will propagate through the entire solution. Therefore, double-checking the substitution is always a good practice. The quadratic formula, when applied correctly, leads directly to the solutions of the quadratic equation.

Simplifying the expression under the square root, we have $144 + 216 = 360$. So, the equation becomes: $x = \\frac12 \\pm \\sqrt{360}}{2}$. Now we need to simplify the square root of 360. We look for the largest perfect square that divides 360. Since $360 = 36 imes 10$, and 36 is a perfect square, we can rewrite $\\\\sqrt{360}$ as $\\\\sqrt{36 imes 10} = \\\\sqrt{36} imes \\\\sqrt{10} = 6\\\\sqrt{10}$. Substituting this back into the equation, we get $x = \\frac{12 \\pm 6\\sqrt{10}{2}$. This simplification is crucial for expressing the solution in its simplest form. Identifying and factoring out perfect squares from under the square root is a common technique in algebra. The next step involves simplifying the entire expression by dividing both terms in the numerator by the denominator.

Dividing both terms in the numerator by 2, we get: $x = \\frac12}{2} \\pm \\frac{6\\sqrt{10}}{2} = 6 \\pm 3\\sqrt{10}$. This gives us the same two solutions as before $x = 6 + 3\\sqrt{10$ and $x = 6 - 3\\\\sqrt{10}$. These are the values of $x$ that satisfy the original quadratic equation. The final simplification step ensures that the solution is presented in its most concise form. The quadratic formula, when correctly applied and simplified, provides a reliable method for solving quadratic equations. This method is particularly useful when the equation is difficult to factor or when completing the square is cumbersome. By arriving at the same solutions as the completing the square method, we have validated our results and demonstrated the consistency of different approaches to solving quadratic equations.

Conclusion

In conclusion, we have successfully solved the quadratic equation $x^2 - 12x + 36 = 90$ using two different methods: completing the square and the quadratic formula. Both methods led us to the same solutions: $x = 6 + 3\\\\sqrt{10}$ and $x = 6 - 3\\\\sqrt{10}$. This demonstrates the versatility and reliability of these techniques in solving quadratic equations. Understanding these methods is essential for anyone studying algebra and related fields. The ability to solve quadratic equations is not only a fundamental mathematical skill but also a crucial tool in various real-world applications. Whether you prefer the step-by-step approach of completing the square or the directness of the quadratic formula, having both methods in your toolkit will enhance your problem-solving abilities. The process of solving quadratic equations reinforces key algebraic concepts and techniques, contributing to a deeper understanding of mathematical principles. By mastering these methods, you will be well-prepared to tackle more complex mathematical problems in the future.

The correct answer is A. $x = 6 \\\\pm 3 \\\\sqrt{10}$. We have demonstrated two methods to arrive at this solution, providing a comprehensive understanding of the problem-solving process. Remember to always double-check your work and consider different approaches to ensure accuracy and deepen your understanding.