Solving X² - 10x = -34 A Step-by-Step Guide To Quadratic Equations

In the realm of mathematics, quadratic equations hold a significant position, serving as fundamental tools for modeling and solving a myriad of real-world problems. These equations, characterized by their second-degree polynomial form, present intriguing challenges and offer elegant solutions. In this comprehensive guide, we delve into the intricacies of solving quadratic equations, focusing on the specific example of x² - 10x = -34. We will explore various methods, dissect the solutions, and provide a clear understanding of the concepts involved.

Quadratic equations are equations of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation, also known as roots or zeros, represent the values of x that satisfy the equation. These solutions can be real or complex numbers, depending on the nature of the equation's coefficients.

Solving quadratic equations involves finding these roots, and several methods are available to accomplish this. Among the most common techniques are factoring, completing the square, and the quadratic formula. Each method possesses its strengths and weaknesses, and the choice of method often depends on the specific form of the equation and the solver's preference.

Our focus equation is x² - 10x = -34. This equation, while seemingly simple, presents a unique challenge that necessitates the application of specific techniques to arrive at the correct solutions. The key to solving this equation lies in recognizing its quadratic nature and employing appropriate methods to isolate the variable x.

Before diving into the solution process, let's rewrite the equation in the standard quadratic form: ax² + bx + c = 0. By adding 34 to both sides of the equation, we obtain:

x² - 10x + 34 = 0

Now, the equation is in a form that readily lends itself to various solution methods. We can proceed with techniques such as completing the square or the quadratic formula to find the values of x that satisfy this equation.

Completing the square is a powerful technique for solving quadratic equations, especially when factoring proves difficult. This method involves manipulating the equation to create a perfect square trinomial on one side, which can then be easily factored.

To complete the square for the equation x² - 10x + 34 = 0, we follow these steps:

1. Isolate the quadratic and linear terms: Subtract 34 from both sides to get x² - 10x = -34.
2. Complete the square: Take half of the coefficient of the x term (-10), square it ((-10/2)² = 25), and add it to both sides of the equation. This gives us x² - 10x + 25 = -34 + 25.
3. Factor the perfect square trinomial: The left side of the equation can now be factored as (x - 5)² = -9.
4. Take the square root of both sides: √(x - 5)² = ±√-9, which simplifies to x - 5 = ±3i, where i is the imaginary unit (√-1).
5. Solve for x: Add 5 to both sides to get x = 5 ± 3i.

Thus, the solutions to the equation x² - 10x = -34 obtained by completing the square are x = 5 + 3i and x = 5 - 3i. These solutions are complex numbers, indicating that the quadratic equation has no real roots.

The quadratic formula is a versatile tool that provides a direct solution for any quadratic equation. It is derived from the process of completing the square and offers a systematic approach to finding the roots.

The quadratic formula states that for an equation of the form ax² + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b² - 4ac)) / 2a

Applying the quadratic formula to our equation x² - 10x + 34 = 0, we identify the coefficients as a = 1, b = -10, and c = 34. Substituting these values into the formula, we get:

x = (10 ± √((-10)² - 4 * 1 * 34)) / (2 * 1)

Simplifying the expression under the square root:

x = (10 ± √(100 - 136)) / 2

x = (10 ± √-36) / 2

Since the discriminant (the value under the square root) is negative, the solutions will be complex numbers:

x = (10 ± 6i) / 2

Dividing both terms in the numerator by 2:

x = 5 ± 3i

Therefore, the solutions to the equation x² - 10x = -34 obtained using the quadratic formula are x = 5 + 3i and x = 5 - 3i, which are consistent with the solutions obtained by completing the square.

The solutions to the equation x² - 10x = -34 are x = 5 + 3i and x = 5 - 3i. These solutions are complex numbers, which means they have a real part (5) and an imaginary part (3i and -3i). Complex roots arise when the discriminant (b² - 4ac) of the quadratic equation is negative.

The presence of complex roots indicates that the parabola represented by the quadratic equation does not intersect the x-axis. In other words, there are no real values of x for which the equation equals zero. This understanding is crucial in various applications, such as electrical engineering and quantum mechanics, where complex numbers are used to model oscillatory behavior and wave phenomena.

Both completing the square and the quadratic formula are effective methods for solving quadratic equations, but the choice of method often depends on the specific equation and the solver's preference. Completing the square is particularly useful when the coefficient of the x² term is 1 and the coefficient of the x term is an even number. This method provides a deeper understanding of the structure of the quadratic equation and its relationship to perfect square trinomials.

The quadratic formula, on the other hand, is a more general method that can be applied to any quadratic equation, regardless of the coefficients. It is especially advantageous when the equation is not easily factorable or when the coefficients are complex or irrational numbers. The quadratic formula guarantees a solution, even when completing the square may be cumbersome.

In the case of the equation x² - 10x = -34, both methods are equally applicable. Completing the square provides a step-by-step approach that reveals the underlying structure of the equation, while the quadratic formula offers a direct and efficient solution.

Solving quadratic equations is a fundamental skill in mathematics with wide-ranging applications in various fields. By mastering techniques such as completing the square and the quadratic formula, we gain the ability to tackle a variety of problems involving quadratic relationships.

In this guide, we have explored the process of solving the quadratic equation x² - 10x = -34. We have demonstrated the application of both completing the square and the quadratic formula, arriving at the complex solutions x = 5 + 3i and x = 5 - 3i. These solutions highlight the importance of understanding complex numbers and their role in quadratic equations.

By understanding the concepts and methods presented in this guide, you can confidently approach and solve a wide range of quadratic equations, expanding your mathematical toolkit and enhancing your problem-solving abilities.

Final Answer: The correct answer is B. x = 5 ± 3i