Solving X In Quadratic Equation X² - 12x + 59 = 0
In this comprehensive article, we will delve into the process of solving for x in the quadratic equation x² - 12x + 59 = 0. Quadratic equations, characterized by the general form ax² + bx + c = 0, are fundamental in mathematics and find applications in various fields, including physics, engineering, and computer science. Mastering the techniques to solve these equations is crucial for anyone pursuing studies or careers in these areas. We will explore the quadratic formula, a powerful tool for finding the roots (or solutions) of any quadratic equation, regardless of whether the roots are real or complex. By meticulously applying this formula, we will determine the values of x that satisfy the given equation. Furthermore, we will discuss the nature of the roots, distinguishing between real and complex solutions, and interpreting their significance within the context of the equation.
Before we dive into the solution, it's essential to understand the basics of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (x in this case) is 2. The general form of a quadratic equation is:
ax² + bx + c = 0
where a, b, and c are constants, and a ≠ 0. The coefficients a, b, and c play a crucial role in determining the nature and values of the solutions (also known as roots) of the equation. The solutions are the values of x that make the equation true. These roots can be real numbers, which can be plotted on a number line, or complex numbers, which involve the imaginary unit i (where i² = -1).
In our given equation, x² - 12x + 59 = 0, we can identify the coefficients as follows:
- a = 1
- b = -12
- c = 59
These coefficients will be instrumental in applying the quadratic formula to find the solutions for x. The quadratic formula is derived from the process of completing the square and provides a direct method for solving quadratic equations, irrespective of the complexity of the coefficients. It is a cornerstone of algebra and a vital tool for anyone working with polynomial equations.
The quadratic formula is a powerful tool used to find the solutions (roots) of any quadratic equation in the form ax² + bx + c = 0. The formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
This formula provides two possible solutions for x, one using the plus sign (+) and the other using the minus sign (±). The expression inside the square root, b² - 4ac, is known as the discriminant. The discriminant plays a crucial role in determining the nature of the roots:
- If b² - 4ac > 0, the equation has two distinct real roots.
- If b² - 4ac = 0, the equation has one real root (a repeated root).
- If b² - 4ac < 0, the equation has two complex roots (conjugate pairs).
The complex roots involve the imaginary unit i, where i² = -1. These roots are expressed in the form p + qi and p - qi, where p and q are real numbers. Understanding the discriminant allows us to predict the type of solutions we will obtain before even applying the full quadratic formula, saving time and providing valuable insight into the equation's behavior. In our case, we will use this formula to solve for x in the equation x² - 12x + 59 = 0.
Now, let's apply the quadratic formula to our equation, x² - 12x + 59 = 0. We have already identified the coefficients:
- a = 1
- b = -12
- c = 59
Substitute these values into the quadratic formula:
x = (-(-12) ± √((-12)² - 4 * 1 * 59)) / (2 * 1)
Simplify the expression step by step:
x = (12 ± √(144 - 236)) / 2
x = (12 ± √(-92)) / 2
Since the value inside the square root is negative, we have complex roots. We can rewrite √(-92) using the imaginary unit i:
√(-92) = √(92 * -1) = √92 * √-1 = √92 * i
We can further simplify √92 by factoring out the largest perfect square:
√92 = √(4 * 23) = √4 * √23 = 2√23
So, √(-92) = 2√23 * i
Substitute this back into our equation:
x = (12 ± 2√23 * i) / 2
Now, divide both terms in the numerator by 2:
x = 6 ± √23 * i
Thus, the solutions for x are 6 + √23 * i and 6 - √23 * i. These are complex conjugate roots, as expected due to the negative discriminant. The meticulous application of the quadratic formula, combined with careful simplification of the resulting expressions, has allowed us to find the exact solutions for x in this quadratic equation.
The solutions for the equation x² - 12x + 59 = 0 are:
x = 6 + √23 * i
and
x = 6 - √23 * i
These are complex conjugate roots, which means they are of the form p + qi and p - qi, where p and q are real numbers and i is the imaginary unit. In this case, p = 6 and q = √23. Complex roots occur when the discriminant (b² - 4ac) is negative, indicating that the quadratic equation has no real solutions. The complex solutions represent points that do not intersect the x-axis when the quadratic equation is graphed as a parabola. Understanding the nature of the roots is crucial in various mathematical and engineering applications, as it provides insights into the behavior of the system or function being modeled by the equation. The solutions we have found are precise and accurately represent the values of x that satisfy the given quadratic equation.
In conclusion, we have successfully solved for x in the quadratic equation x² - 12x + 59 = 0 using the quadratic formula. The solutions are complex conjugates:
x = 6 ± √23 * i
This exercise demonstrates the power and versatility of the quadratic formula in solving quadratic equations, regardless of whether the roots are real or complex. By carefully applying the formula and simplifying the resulting expressions, we can find the exact values of x that satisfy the equation. The nature of the roots, determined by the discriminant, provides valuable information about the equation's behavior and the characteristics of its solutions. Mastering the techniques for solving quadratic equations is essential for anyone studying mathematics, science, or engineering, as these equations appear in numerous applications and contexts. The ability to confidently and accurately solve these equations is a testament to one's mathematical proficiency and problem-solving skills. This detailed walkthrough has provided a clear and comprehensive understanding of the process, ensuring that readers can apply these techniques to solve other quadratic equations they encounter.
