Solving X² - 12x + 59 = 0 A Step By Step Guide
This article delves into solving the quadratic equation x² - 12x + 59 = 0. Quadratic equations, a fundamental concept in mathematics, appear in various fields, including physics, engineering, and computer science. Mastering the techniques to solve them is crucial for anyone pursuing studies or careers in these areas. We will explore the use of the quadratic formula, a powerful tool for finding the roots (or solutions) of any quadratic equation in the standard form ax² + bx + c = 0. By understanding the step-by-step process, readers will gain a solid foundation for tackling similar problems and deepen their appreciation for the elegance of algebraic solutions.
The quadratic formula is a direct method derived from the process of completing the square, providing a universal solution for any quadratic equation. It states that for an equation of the form ax² + bx + c = 0, the solutions for x are given by: x = (-b ± √(b² - 4ac)) / 2a. This formula elegantly encapsulates the relationship between the coefficients of the quadratic equation and its roots. The term inside the square root, b² - 4ac, is known as the discriminant, which plays a crucial role in determining the nature of the roots. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real root (a repeated root), and a negative discriminant indicates two complex roots. Understanding the discriminant is key to interpreting the solutions obtained from the quadratic formula and gaining deeper insights into the behavior of quadratic equations.
In this specific case, the equation x² - 12x + 59 = 0 has coefficients a = 1, b = -12, and c = 59. By carefully substituting these values into the quadratic formula, we can systematically find the values of x that satisfy the equation. The process involves careful arithmetic and algebraic manipulation, ensuring that each step is accurately executed. The ± sign in the formula signifies that there are generally two solutions, corresponding to the addition and subtraction operations. These solutions represent the points where the parabola defined by the quadratic equation intersects the x-axis (in the case of real roots) or provide complex solutions if the parabola does not intersect the x-axis. This exploration will not only provide the answer to this specific problem but will also solidify the reader's understanding of the quadratic formula and its applications.
Applying the Quadratic Formula
To solve for x in the equation x² - 12x + 59 = 0, we will utilize the quadratic formula. As previously mentioned, the quadratic formula is given by x = (-b ± √(b² - 4ac)) / 2a. This formula is applicable to any quadratic equation in the standard form ax² + bx + c = 0. The power of this formula lies in its generality; regardless of the specific values of the coefficients a, b, and c, the quadratic formula provides a direct route to finding the solutions. Its derivation through the method of completing the square underscores its mathematical rigor and provides a deeper understanding of its structure. In essence, the quadratic formula is a condensed representation of the steps involved in completing the square, making it a highly efficient tool for solving quadratic equations.
Identifying the coefficients is the first crucial step in applying the quadratic formula. In our equation, x² - 12x + 59 = 0, we can clearly identify that a = 1 (the coefficient of x²), b = -12 (the coefficient of x), and c = 59 (the constant term). These coefficients are the numerical anchors that link the equation to the quadratic formula. A common error is to misidentify the coefficients, particularly the sign of b and c. Therefore, meticulous attention to detail in this step is crucial for obtaining the correct solutions. Once the coefficients are correctly identified, they can be systematically substituted into the quadratic formula.
Substituting the values a = 1, b = -12, and c = 59 into the quadratic formula, we get: x = (-(-12) ± √((-12)² - 4 * 1 * 59)) / (2 * 1). This substitution is the mechanical application of the formula, but it's where careful arithmetic begins. The negative signs must be handled carefully, and the order of operations (PEMDAS/BODMAS) must be strictly followed. The expression within the square root represents the discriminant, which will determine the nature of the roots. The denominator is simply 2a, which in this case is 2 * 1 = 2. The next step involves simplifying this expression, which will lead us to the solutions for x. This process of substitution and simplification demonstrates the elegance and power of the quadratic formula in action.
Simplifying the Expression
Following the substitution, we now focus on simplifying the expression obtained from the quadratic formula: x = (12 ± √(144 - 236)) / 2. The simplification process begins within the square root. We first calculate (-12)² which equals 144. Then, we calculate 4 * 1 * 59, which equals 236. Subtracting 236 from 144 gives us -92. The presence of a negative number under the square root indicates that the solutions will be complex numbers, a key insight into the nature of the roots. This is a direct consequence of the discriminant (b² - 4ac) being negative. Understanding the implications of the discriminant allows us to predict the type of solutions we will obtain even before fully solving the equation.
The next step involves dealing with the square root of a negative number. Recall that the square root of -1 is defined as the imaginary unit 'i'. Therefore, we can rewrite √(-92) as √(92 * -1) = √(92) * √(-1) = √(92) * i. This step introduces the concept of imaginary numbers, which are essential for solving quadratic equations with negative discriminants. The ability to manipulate imaginary numbers is crucial for working with complex solutions. Further simplification of √(92) is possible by factoring out perfect squares. We can express 92 as 4 * 23, so √(92) = √(4 * 23) = √4 * √23 = 2√23. This simplification allows us to express the solution in its most reduced form.
Substituting this back into the expression, we get x = (12 ± 2√23 * i) / 2. The final step in simplification involves dividing both terms in the numerator by the denominator. Dividing 12 by 2 gives us 6, and dividing 2√23 * i by 2 gives us √23 * i. Therefore, the simplified solutions are x = 6 ± √23 * i. These are the complex roots of the given quadratic equation. The solution consists of a real part (6) and an imaginary part (√23 * i). This detailed step-by-step simplification process illustrates how the quadratic formula leads to the solutions, even when they are complex numbers.
The Solutions
After simplifying the expression, we arrive at the solutions for x: x = 6 ± √23 * i. These solutions are complex numbers, consisting of a real part (6) and an imaginary part (√23 * i). The presence of the imaginary unit 'i' signifies that the quadratic equation has no real roots; in other words, the parabola represented by the equation does not intersect the x-axis. Complex numbers are a fundamental concept in mathematics, extending the number system beyond real numbers and allowing us to solve equations that would otherwise have no solutions. Understanding complex numbers is essential for various advanced topics in mathematics, physics, and engineering.
The two solutions are x = 6 + √23 * i and x = 6 - √23 * i. These solutions are complex conjugates of each other. Complex conjugate pairs are a common occurrence when solving quadratic equations with negative discriminants. The complex conjugate of a complex number a + bi is a - bi, and vice versa. This symmetry in the solutions is a direct consequence of the quadratic formula and the properties of complex numbers. The fact that the solutions are complex conjugates provides further insight into the nature of the roots and the underlying mathematical structure.
Therefore, the correct answer is C: x = 6 ± √23 * i. This solution demonstrates the power and elegance of the quadratic formula in solving quadratic equations, even when the roots are complex. By meticulously applying the formula and simplifying the resulting expression, we can successfully find the solutions. This process not only provides the answer to the specific problem but also reinforces the understanding of quadratic equations, the quadratic formula, and the concept of complex numbers. Mastering these concepts is crucial for anyone pursuing further studies in mathematics and related fields.
Conclusion
In conclusion, we have successfully solved for x in the quadratic equation x² - 12x + 59 = 0 using the quadratic formula. The solutions are x = 6 ± √23 * i, which are complex numbers. This exercise highlights the importance of the quadratic formula as a versatile tool for finding the roots of any quadratic equation. The process involved identifying the coefficients, substituting them into the formula, simplifying the expression, and dealing with complex numbers. Each step requires careful attention to detail and a solid understanding of algebraic principles.
The discriminant (b² - 4ac) played a crucial role in determining the nature of the roots. In this case, the negative discriminant indicated that the solutions would be complex numbers. This understanding allows us to predict the type of solutions we should expect before fully solving the equation. Complex numbers are a fundamental concept in mathematics, and their appearance in the solutions underscores the richness and complexity of the number system.
This problem serves as a valuable example for students learning about quadratic equations and the quadratic formula. By working through this example, readers can gain confidence in applying the formula and solving similar problems. The step-by-step approach presented in this article provides a clear and concise guide for understanding the process. Furthermore, the discussion of complex numbers and the discriminant enhances the overall understanding of quadratic equations and their solutions. Mastering these concepts is essential for success in mathematics and related fields.
