Solving Quadratic Equations Finding Solutions For X^2 - 6x = -58
In the realm of mathematics, quadratic equations hold a prominent position, serving as fundamental tools for modeling various real-world phenomena. From projectile motion to financial modeling, these equations provide a powerful framework for understanding and predicting complex systems. At its core, a quadratic equation is a polynomial equation of the second degree, characterized by the presence of a squared term. The general form of a quadratic equation is expressed as ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠0. The solutions to a quadratic equation, also known as its roots or zeros, represent the values of the variable x that satisfy the equation. These solutions can be real or complex numbers, depending on the nature of the coefficients a, b, and c. The quest to find these solutions has captivated mathematicians for centuries, leading to the development of various techniques and methods.
The quadratic formula stands as a cornerstone in solving quadratic equations, providing a universal approach to determine the roots regardless of the equation's specific form. This formula, derived from completing the square, expresses the solutions x in terms of the coefficients a, b, and c: x = (-b ± √(b^2 - 4ac)) / 2a. The discriminant, denoted as Δ = b^2 - 4ac, plays a crucial role in determining the nature of the solutions. When Δ > 0, the equation has two distinct real roots, indicating two points where the parabola intersects the x-axis. If Δ = 0, the equation possesses one real root (a repeated root), signifying that the parabola touches the x-axis at a single point. And when Δ < 0, the equation has two complex roots, implying that the parabola does not intersect the x-axis in the real number plane. Understanding the discriminant allows us to anticipate the type of solutions we'll encounter, guiding our problem-solving strategy.
Completing the square, an alternative method for solving quadratic equations, involves manipulating the equation to create a perfect square trinomial. This technique provides a deeper understanding of the structure of quadratic equations and offers a pathway to solve them without relying solely on the quadratic formula. By adding and subtracting (b/2a)^2 to the equation, we transform it into the form (x + b/2a)^2 = (b^2 - 4ac) / 4a^2. Taking the square root of both sides allows us to isolate x and determine the solutions. Completing the square not only solves quadratic equations but also reveals the vertex form of the parabola, providing insights into its symmetry and extreme points. This method is particularly valuable when the quadratic equation does not readily factor or when we seek to understand the geometric properties of the parabola.
Factoring, when applicable, offers the most straightforward approach to solving quadratic equations. This method relies on expressing the quadratic expression as a product of two linear factors. If we can find two numbers that multiply to ac and add up to b, we can rewrite the middle term bx and factor the expression by grouping. For instance, in the equation x^2 + 5x + 6 = 0, we seek two numbers that multiply to 6 and add up to 5. These numbers are 2 and 3, allowing us to rewrite the equation as (x + 2)(x + 3) = 0. Setting each factor to zero, we find the solutions x = -2 and x = -3. Factoring is efficient when the roots are integers or simple fractions, providing a quick and elegant solution. However, not all quadratic equations can be factored easily, making the quadratic formula and completing the square essential tools in our problem-solving arsenal.
Solving quadratic equations can be tricky, but with the right approach, it becomes a manageable task. Let's delve into the specific equation at hand: x^2 - 6x = -58. Our mission is to find the values of x that satisfy this equation. To do so, we'll employ a systematic approach, transforming the equation into a standard form and then applying the powerful quadratic formula.
First, we need to rewrite the equation in the standard quadratic form: ax^2 + bx + c = 0. This involves moving the constant term to the left side of the equation. By adding 58 to both sides, we get: x^2 - 6x + 58 = 0. Now, we have a clear view of the coefficients: a = 1, b = -6, and c = 58. These coefficients are the keys to unlocking the solutions using the quadratic formula.
The quadratic formula, a cornerstone of algebra, provides a universal method for solving quadratic equations. It states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by: x = (-b ± √(b^2 - 4ac)) / 2a. This formula might seem daunting at first, but with practice, it becomes a reliable tool. Now, let's substitute the values of a, b, and c that we identified earlier into this formula.
Substituting a = 1, b = -6, and c = 58, we get: x = (6 ± √((-6)^2 - 4 * 1 * 58)) / (2 * 1). Let's break this down step by step. First, we calculate the discriminant, which is the expression under the square root: (-6)^2 - 4 * 1 * 58 = 36 - 232 = -196. The discriminant, -196, is a negative number. This tells us that the solutions to this equation will be complex numbers. Complex numbers involve the imaginary unit 'i', where i^2 = -1. This is a crucial piece of information that guides our next steps.
Now, we can simplify the square root of the negative discriminant: √(-196) = √(196 * -1) = √196 * √-1 = 14i. This simplification allows us to express the solutions in terms of the imaginary unit. Substituting this back into the quadratic formula, we get: x = (6 ± 14i) / 2. We can further simplify this by dividing both the real and imaginary parts by 2: x = 3 ± 7i. This gives us two complex solutions: x = 3 + 7i and x = 3 - 7i. These are the values of x that satisfy the original equation.
Therefore, the solutions to the quadratic equation x^2 - 6x = -58 are x = 3 + 7i and x = 3 - 7i. These solutions are complex conjugates, meaning they have the same real part but opposite imaginary parts. This is a common occurrence when solving quadratic equations with negative discriminants. By following a systematic approach and applying the quadratic formula, we successfully navigated the complexities of this equation and arrived at the correct solutions.
Understanding complex solutions is crucial when dealing with quadratic equations that have a negative discriminant. In the previous section, we solved the equation x^2 - 6x = -58 and found the solutions to be x = 3 + 7i and x = 3 - 7i. Now, let's delve deeper into the structure of these solutions and identify the values of 'a' and 'b' in the general form of complex solutions: x = a + bi and x = a - bi. This will solidify our understanding of complex numbers and their representation.
The general form x = a + bi represents a complex number, where 'a' is the real part and 'b' is the imaginary part. The imaginary part is multiplied by the imaginary unit 'i', where i^2 = -1. The conjugate of a complex number a + bi is a - bi, which has the same real part but the opposite imaginary part. When solving quadratic equations with a negative discriminant, the solutions often come in the form of complex conjugate pairs. This is a direct consequence of the quadratic formula and the nature of square roots of negative numbers.
In our specific case, we have the solutions x = 3 + 7i and x = 3 - 7i. Comparing these solutions with the general form x = a + bi and x = a - bi, we can directly identify the values of 'a' and 'b'. For both solutions, the real part is 3. Therefore, a = 3. The imaginary part in the solution x = 3 + 7i is 7, and in the solution x = 3 - 7i, it is -7. However, in the general form, 'b' represents the magnitude of the imaginary part. Therefore, b = 7. It's important to note that 'b' is a positive value, representing the coefficient of the imaginary unit 'i'.
So, in the solutions x = 3 + 7i and x = 3 - 7i, we have identified a = 3 and b = 7. These values completely define the complex solutions to the quadratic equation x^2 - 6x = -58. The real part 'a' represents the position of the complex number on the real number line, while the imaginary part 'b' represents its position on the imaginary number line. Together, 'a' and 'b' uniquely locate the complex number in the complex plane.
By understanding the structure of complex solutions and correctly identifying 'a' and 'b', we gain a deeper appreciation for the nature of quadratic equations and their solutions. This knowledge is essential for further exploration of complex numbers and their applications in various fields of mathematics, physics, and engineering. The ability to decompose complex solutions into their real and imaginary parts is a fundamental skill in working with these numbers and solving related problems.
Solving quadratic equations is a fundamental skill in mathematics with widespread applications across various disciplines. From physics and engineering to economics and finance, quadratic equations provide a powerful tool for modeling and understanding real-world phenomena. In this comprehensive guide, we've explored the core concepts and techniques for solving quadratic equations, empowering you to tackle these equations with confidence and precision.
We began by defining the general form of a quadratic equation: ax^2 + bx + c = 0, where a, b, and c are constants and a ≠0. We then delved into the various methods for finding the solutions, also known as roots or zeros, of these equations. The quadratic formula, a cornerstone of algebra, provides a universal solution, expressing the roots in terms of the coefficients a, b, and c: x = (-b ± √(b^2 - 4ac)) / 2a. We emphasized the importance of the discriminant, Δ = b^2 - 4ac, in determining the nature of the solutions: two distinct real roots when Δ > 0, one real root when Δ = 0, and two complex roots when Δ < 0.
We also explored completing the square, an alternative method that not only solves quadratic equations but also reveals the vertex form of the parabola, providing insights into its symmetry and extreme points. Factoring, when applicable, offers the most straightforward approach, allowing us to express the quadratic expression as a product of two linear factors. However, we acknowledged that not all quadratic equations can be factored easily, making the quadratic formula and completing the square essential tools in our problem-solving arsenal.
We then tackled the specific equation x^2 - 6x = -58, demonstrating a step-by-step solution using the quadratic formula. We highlighted the importance of rewriting the equation in standard form, identifying the coefficients, and carefully substituting them into the formula. The negative discriminant led us to complex solutions, involving the imaginary unit 'i'. We simplified the square root of the negative discriminant and expressed the solutions in the form x = a ± bi, where 'a' is the real part and 'b' is the imaginary part.
Finally, we focused on identifying the values of 'a' and 'b' in the complex solutions x = 3 + 7i and x = 3 - 7i. We emphasized the general form of complex numbers, a + bi, and their conjugates, a - bi. By comparing the solutions with the general form, we readily identified a = 3 and b = 7. This exercise solidified our understanding of complex numbers and their representation in the complex plane.
By mastering the concepts and techniques presented in this guide, you'll be well-equipped to solve a wide range of quadratic equations, whether they have real or complex solutions. Remember to approach each equation systematically, choose the most appropriate method, and carefully execute each step. With practice and perseverance, you'll unlock the power of quadratic equations and their applications in the world around us.
