Solving Z² - 16z + 30 = -74 A Step By Step Guide
In this article, we will delve into the process of solving the quadratic equation z² - 16z + 30 = -74. Quadratic equations, characterized by the highest power of the variable being 2, are fundamental in mathematics and have wide-ranging applications in various fields such as physics, engineering, and economics. Our goal is to find the values of 'z' that satisfy the given equation. This will involve manipulating the equation, employing the quadratic formula, and simplifying the solutions to their fullest extent, including addressing non-real solutions. By the end of this article, you will have a clear understanding of how to approach and solve quadratic equations of this nature.
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable. The solutions to a quadratic equation, also known as roots or zeros, are the values of 'x' that make the equation true. These solutions can be real numbers, complex numbers, or both. The quadratic formula is a powerful tool used to find these solutions. It is derived by completing the square on the general form of the quadratic equation and is given by:
x = (-b ± √(b² - 4ac)) / 2a
where 'a', 'b', and 'c' are the coefficients of the quadratic equation. The term b² - 4ac is known as the discriminant, which provides valuable information about the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has one real root (a repeated root). If it is negative, the equation has two complex conjugate roots. Understanding these concepts is crucial for solving and interpreting the solutions of quadratic equations.
Transforming the Equation
The initial step in solving the given equation, z² - 16z + 30 = -74, is to transform it into the standard quadratic form ax² + bx + c = 0. This is achieved by moving all terms to one side of the equation, leaving zero on the other side. In this case, we need to add 74 to both sides of the equation. This results in:
z² - 16z + 30 + 74 = 0
Simplifying the equation by combining the constant terms, we get:
z² - 16z + 104 = 0
Now, the equation is in the standard quadratic form, where a = 1, b = -16, and c = 104. This form allows us to readily apply the quadratic formula to find the solutions for 'z'. The transformation process is a crucial step as it sets the stage for the subsequent application of the quadratic formula and ensures accurate solutions.
Applying the Quadratic Formula
With the equation now in the standard form z² - 16z + 104 = 0, we can apply the quadratic formula to find the solutions for 'z'. The quadratic formula, as mentioned earlier, is:
z = (-b ± √(b² - 4ac)) / 2a
In our equation, a = 1, b = -16, and c = 104. Substituting these values into the formula, we get:
z = (-(-16) ± √((-16)² - 4 * 1 * 104)) / (2 * 1)
Simplifying the expression step by step:
z = (16 ± √(256 - 416)) / 2
z = (16 ± √(-160)) / 2
The presence of a negative number under the square root indicates that the solutions will be complex numbers. This is a common occurrence in quadratic equations and highlights the importance of understanding complex numbers in mathematics. The next step involves simplifying the square root of the negative number.
Simplifying the Solutions
We've arrived at the expression z = (16 ± √(-160)) / 2. To simplify this further, we need to address the square root of -160. Recall that the square root of -1 is denoted by 'i', the imaginary unit. We can rewrite √(-160) as √(160 * -1) = √(160) * √(-1) = √(160) * i. Now we need to simplify √(160).
First, find the prime factorization of 160: 160 = 2 * 80 = 2 * 2 * 40 = 2 * 2 * 2 * 20 = 2 * 2 * 2 * 2 * 10 = 2 * 2 * 2 * 2 * 2 * 5 = 2⁵ * 5. Therefore, √(160) = √(2⁵ * 5) = √(2⁴ * 2 * 5) = 2²√(2 * 5) = 4√10.
Substituting this back into our expression, we have:
z = (16 ± 4√10 * i) / 2
Now, we can divide both terms in the numerator by 2:
z = 8 ± 2√10 * i
This gives us two complex solutions: z = 8 + 2√10 * i and z = 8 - 2√10 * i.
Final Solutions
After applying the quadratic formula and simplifying the results, we have found the solutions to the equation z² - 16z + 30 = -74. The solutions are complex numbers, specifically:
z = 8 + 2√10 * i and z = 8 - 2√10 * i
These solutions represent the values of 'z' that satisfy the original equation. Complex solutions often arise in quadratic equations when the discriminant (b² - 4ac) is negative, as is the case in this problem. Understanding how to work with complex numbers is essential for solving a wide range of mathematical problems. These solutions are fully simplified and represent the final answer to the problem.
In conclusion, we have successfully solved the quadratic equation z² - 16z + 30 = -74 by transforming it into standard form, applying the quadratic formula, and simplifying the solutions. We encountered complex solutions due to the negative discriminant, highlighting the importance of complex numbers in solving quadratic equations. The solutions we found are z = 8 + 2√10 * i and z = 8 - 2√10 * i. This process demonstrates a comprehensive approach to solving quadratic equations, applicable to various problems in mathematics and related fields. Understanding these steps and concepts is crucial for anyone studying algebra and beyond. The ability to solve quadratic equations is a fundamental skill that opens doors to more advanced mathematical topics and real-world applications.
