Solving $x^2 - 12x + 59 = 0$ A Step-by-Step Guide
Solving quadratic equations is a fundamental concept in algebra. Quadratic equations, characterized by the general form ax² + bx + c = 0, appear in various fields of mathematics, physics, engineering, and economics. This article provides a detailed step-by-step solution for finding the values of x that satisfy the quadratic equation x² - 12x + 59 = 0. We will explore the quadratic formula, a powerful tool for solving these types of equations, and demonstrate its application in this specific case. By understanding the process, readers will gain a solid foundation for tackling similar problems and appreciating the significance of quadratic equations in different contexts. This article aims to provide a comprehensive explanation that not only solves the problem but also enhances the reader's understanding of the underlying principles. Let's begin by examining the equation and identifying the coefficients required for the quadratic formula.
To solve the quadratic equation x² - 12x + 59 = 0, we first need to identify the coefficients a, b, and c. In the general form of a quadratic equation, ax² + bx + c = 0, a represents the coefficient of the x² term, b represents the coefficient of the x term, and c represents the constant term. In our equation, x² - 12x + 59 = 0, the coefficients are as follows:
- a = 1 (the coefficient of x²)
- b = -12 (the coefficient of x)
- c = 59 (the constant term)
These coefficients are crucial for applying the quadratic formula, which is a reliable method for finding the roots (or solutions) of any quadratic equation. The quadratic formula is derived from the process of completing the square and provides a direct way to calculate the values of x that satisfy the equation. Understanding these coefficients and their roles is the first step toward solving the equation effectively. With these values identified, we can now proceed to apply the quadratic formula and determine the solutions for x. This methodical approach ensures accuracy and clarity in the solving process.
Now that we have identified the coefficients, we can apply the quadratic formula. The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
This formula provides the solutions for x in the quadratic equation ax² + bx + c = 0. By substituting the values of a, b, and c that we identified earlier, we can find the solutions for our specific equation, x² - 12x + 59 = 0. Let's substitute the values:
- a = 1
- b = -12
- c = 59
Plugging these values into the formula, we get:
x = (-(-12) ± √((-12)² - 4 * 1 * 59)) / (2 * 1)
This substitution sets the stage for the next step, which involves simplifying the expression under the square root and further calculations to find the values of x. The quadratic formula is a powerful tool, but careful substitution and simplification are key to arriving at the correct solutions. By meticulously applying the formula, we can determine the roots of the equation and gain a deeper understanding of its properties. Next, we will simplify the expression and continue our journey towards the final solution.
With the values substituted into the quadratic formula, the next step is to simplify the expression. The formula, as we have it now, is:
x = (12 ± √(144 - 4 * 1 * 59)) / 2
First, let's simplify the expression under the square root. We need to calculate 144 - 4 * 1 * 59. This involves multiplying 4 by 1 and 59, and then subtracting the result from 144. Let's perform the calculation:
4 * 1 * 59 = 236
Now, subtract this from 144:
144 - 236 = -92
So, the expression under the square root simplifies to -92. Now our equation looks like this:
x = (12 ± √(-92)) / 2
The presence of a negative number under the square root indicates that the solutions will be complex numbers. This is an important observation, as it tells us that the roots of the equation are not real numbers but have an imaginary component. Understanding this aspect is crucial for correctly interpreting the solutions. In the next section, we will deal with the square root of a negative number and further simplify the expression to find the complex solutions for x. This step-by-step simplification ensures we handle each part of the equation with precision.
In the previous step, we arrived at the expression x = (12 ± √(-92)) / 2. The presence of √(-92) indicates that we need to deal with the imaginary unit, denoted by i, where i = √(-1). To simplify √(-92), we can rewrite it as √(-1 * 92). Using the property of square roots, we can separate this into √(-1) * √(92). Since √(-1) = i, we now have i√(92).
Next, we need to simplify √(92). To do this, we look for perfect square factors of 92. The prime factorization of 92 is 2 * 2 * 23, which can be written as 2² * 23. Therefore, √(92) = √(2² * 23) = √(2²) * √(23) = 2√(23).
Substituting this back into our expression, we get i√(92) = i * 2√(23) = 2i√(23). Now, our equation becomes:
x = (12 ± 2i√(23)) / 2
This simplification is crucial for expressing the solutions in the standard form of complex numbers, which is a + bi, where a and b are real numbers. By correctly handling the imaginary unit and simplifying the square root, we are closer to finding the final solutions for x. In the next step, we will divide both terms in the numerator by the denominator to obtain the solutions in their simplest form.
Having simplified the expression to x = (12 ± 2i√(23)) / 2, we can now find the solutions for x. To do this, we divide both terms in the numerator by the denominator, which is 2. This gives us:
x = 12/2 ± (2i√(23))/2
Dividing 12 by 2 yields 6, and dividing 2i√(23) by 2 yields i√(23). Therefore, the solutions for x are:
x = 6 ± i√(23)
These are complex solutions, meaning they have both a real part and an imaginary part. The ± sign indicates that there are two solutions: one with addition and one with subtraction. The two solutions are:
- x = 6 + i√(23)
- x = 6 - i√(23)
These solutions demonstrate that the quadratic equation x² - 12x + 59 = 0 has two complex roots. The solutions are in the standard form of complex numbers, a + bi, where a is the real part and b is the imaginary part. This final simplification and presentation of the solutions complete the process of solving the quadratic equation. Understanding the nature of these solutions—being complex in this case—is essential for a comprehensive understanding of quadratic equations.
In conclusion, we have successfully solved for x in the equation x² - 12x + 59 = 0. By identifying the coefficients, applying the quadratic formula, simplifying the expression, dealing with the imaginary unit, and finding the solutions, we determined that the roots of the equation are complex numbers. The solutions are:
- x = 6 + i√(23)
- x = 6 - i√(23)
These solutions can be compactly written as x = 6 ± i√(23). This exercise demonstrates the power and utility of the quadratic formula in solving quadratic equations, even those with complex roots. The step-by-step approach used in this article provides a clear and methodical way to tackle such problems. Understanding the process and the nature of the solutions is crucial for a deeper understanding of algebra and its applications in various fields. Solving quadratic equations is a fundamental skill, and this example illustrates the key steps and considerations involved in the process. The result matches option C, which confirms the correctness of our solution. By mastering these techniques, readers will be well-equipped to handle a wide range of quadratic equations and related problems.
Final Answer: The final answer is (C)
