Solving The Quadratic Equation X² - 10x = -34 A Step-by-Step Guide
In the realm of mathematics, quadratic equations hold a pivotal position, serving as fundamental tools for modeling and solving a myriad of real-world problems. These equations, characterized by their second-degree polynomial nature, often present multiple solutions, also known as roots. The quest to determine these solutions forms a cornerstone of algebraic understanding. In this article, we embark on a comprehensive exploration of the quadratic equation x² - 10x = -34, delving into the techniques and methodologies employed to unravel its solutions. Our journey will encompass a step-by-step analysis, ensuring clarity and fostering a deep appreciation for the underlying mathematical principles. This article will help you select the correct answer.
At its core, a quadratic equation is a mathematical expression that conforms to the general form ax² + bx + c = 0, where 'a', 'b', and 'c' represent constant coefficients, and 'x' symbolizes the unknown variable whose values we seek to determine. The solutions to a quadratic equation, often referred to as roots or zeros, correspond to the values of 'x' that satisfy the equation, effectively making the expression equal to zero. These solutions hold significant importance, as they represent the points where the parabola defined by the quadratic equation intersects the x-axis on a graphical representation.
Several techniques exist for solving quadratic equations, each with its unique strengths and applicability. Among the most prevalent methods are factoring, completing the square, and the quadratic formula. Factoring involves expressing the quadratic equation as a product of two linear factors, thereby allowing us to identify the solutions by setting each factor equal to zero. Completing the square entails manipulating the equation to create a perfect square trinomial, which can then be readily solved by taking the square root of both sides. The quadratic formula, a universal solution, provides a direct means of calculating the solutions, regardless of the equation's factorability. This formula, expressed as x = (-b ± √(b² - 4ac)) / 2a, elegantly captures the relationship between the coefficients and the solutions.
Our focus now shifts to the specific quadratic equation at hand: x² - 10x = -34. To effectively tackle this equation, we must first transform it into the standard quadratic form, ax² + bx + c = 0. This transformation involves transposing the constant term, -34, to the left side of the equation, resulting in x² - 10x + 34 = 0. Now, the equation is poised for the application of our chosen solution techniques. This step is crucial as it sets the stage for applying the quadratic formula or completing the square method.
Completing the square is a powerful technique that allows us to rewrite the quadratic equation in a form that is easily solvable. This method involves manipulating the equation to create a perfect square trinomial on one side. Let's apply this technique to our equation, x² - 10x + 34 = 0. First, we focus on the terms involving 'x', namely x² - 10x. To complete the square, we take half of the coefficient of the 'x' term (-10), which is -5, and square it, resulting in (-5)² = 25. We then add and subtract this value within the equation:
x² - 10x + 25 - 25 + 34 = 0
This manipulation allows us to rewrite the first three terms as a perfect square trinomial:
(x - 5)² - 25 + 34 = 0
Simplifying the equation, we get:
(x - 5)² + 9 = 0
Now, we isolate the squared term:
(x - 5)² = -9
Taking the square root of both sides, we introduce the imaginary unit 'i', where i² = -1:
x - 5 = ±√(-9) = ±√(9 * -1) = ±3i
Finally, we solve for 'x':
x = 5 ± 3i
Therefore, the solutions to the quadratic equation x² - 10x = -34, obtained by completing the square, are x = 5 + 3i and x = 5 - 3i. This method showcases the elegance of algebraic manipulation in arriving at the solution.
The quadratic formula provides a direct and universally applicable method for solving quadratic equations of the form ax² + bx + c = 0. This formula, expressed as x = (-b ± √(b² - 4ac)) / 2a, elegantly relates the solutions to the coefficients of the equation. Let's apply this formula to our equation, x² - 10x + 34 = 0. In this case, a = 1, b = -10, and c = 34. Substituting these values into the quadratic formula, we get:
x = (-(-10) ± √((-10)² - 4 * 1 * 34)) / (2 * 1)
Simplifying the expression:
x = (10 ± √(100 - 136)) / 2
x = (10 ± √(-36)) / 2
Introducing the imaginary unit 'i', where i² = -1:
x = (10 ± √(36 * -1)) / 2
x = (10 ± 6i) / 2
Dividing both terms in the numerator by 2:
x = 5 ± 3i
Thus, the solutions to the quadratic equation x² - 10x = -34, obtained using the quadratic formula, are x = 5 + 3i and x = 5 - 3i. The quadratic formula serves as a powerful tool, especially when factoring or completing the square proves challenging.
The solutions we have obtained, x = 5 + 3i and x = 5 - 3i, are complex numbers. This signifies that the parabola represented by the quadratic equation does not intersect the x-axis in the real number plane. Instead, the solutions lie in the complex plane, where numbers have both real and imaginary components. The discriminant, b² - 4ac, within the quadratic formula provides valuable insight into the nature of the solutions. In our case, the discriminant is (-10)² - 4 * 1 * 34 = -36, a negative value. A negative discriminant indicates that the quadratic equation possesses complex solutions, reinforcing our findings. Understanding the nature of solutions is essential in interpreting the results within the context of the problem.
In this comprehensive exploration, we have successfully unraveled the solutions to the quadratic equation x² - 10x = -34. By employing both the completing the square method and the quadratic formula, we have demonstrated two distinct yet equally effective approaches to solving quadratic equations. Our analysis has revealed that the solutions are complex numbers, x = 5 + 3i and x = 5 - 3i, indicating that the parabola does not intersect the x-axis in the real number plane. This journey has underscored the significance of quadratic equations in mathematics and their ability to model diverse phenomena. The ability to solve quadratic equations is a fundamental skill in mathematics, with applications spanning various fields.
Therefore, the correct answer to the question "What are the solutions of this quadratic equation? " is:
B. x = 5 ± 3i
