Solving X^2 = 5-x Using The Quadratic Formula A Step-by-Step Guide
Introduction
The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. While some quadratic equations can be easily solved by factoring, the quadratic formula provides a universal method that works for all quadratic equations, regardless of whether they have real or complex solutions. In this article, we will explore how to use the quadratic formula to solve the equation x² = 5 - x, providing a step-by-step guide to help you understand the process and arrive at the correct solutions. The quadratic formula is a fundamental concept in algebra and has wide applications in various fields, including physics, engineering, and economics. Mastering the quadratic formula is essential for anyone studying mathematics or related disciplines. This article aims to provide a clear and concise explanation of the quadratic formula and its application, ensuring that readers can confidently solve quadratic equations using this powerful tool.
Understanding the Quadratic Formula
The quadratic formula is derived from the process of completing the square and is given by:
x = (-b ± √(b² - 4ac)) / (2a)
where a, b, and c are the coefficients of the quadratic equation in the standard form ax² + bx + c = 0. The expression inside the square root, b² - 4ac, is called the discriminant. The discriminant plays a crucial role in determining the nature of the solutions:
- If b² - 4ac > 0, the equation has two distinct real solutions.
- If b² - 4ac = 0, the equation has one real solution (a repeated root).
- If b² - 4ac < 0, the equation has two complex solutions.
The quadratic formula is a versatile tool that can be applied to any quadratic equation, regardless of the nature of its solutions. It is particularly useful when factoring is difficult or impossible. Understanding the quadratic formula requires a solid grasp of algebraic manipulations and the concept of square roots. It is also important to be comfortable with the order of operations and the simplification of expressions. The quadratic formula is not just a formula to be memorized; it is a powerful tool that can be used to solve a wide range of problems. By understanding the derivation and application of the quadratic formula, you can gain a deeper understanding of quadratic equations and their solutions. This knowledge will be invaluable as you progress in your mathematical studies.
Step 1: Rewrite the Equation in Standard Form
The first step in using the quadratic formula is to rewrite the given equation in the standard form ax² + bx + c = 0. The given equation is x² = 5 - x. To rewrite this in standard form, we need to move all terms to one side of the equation, leaving zero on the other side. Add x to both sides and subtract 5 from both sides to get:
x² + x - 5 = 0
Now, we can identify the coefficients a, b, and c. In this case:
- a = 1 (the coefficient of x²)
- b = 1 (the coefficient of x)
- c = -5 (the constant term)
Rewriting the equation in standard form is a crucial step because it ensures that we correctly identify the coefficients a, b, and c, which are essential for applying the quadratic formula. A mistake in identifying these coefficients will lead to incorrect solutions. This step also helps us visualize the equation in a familiar format, making it easier to apply the quadratic formula. The standard form allows us to clearly see the relationship between the coefficients and the terms of the equation. This understanding is fundamental to solving quadratic equations and other algebraic problems. By mastering this step, you will be well-prepared to tackle more complex equations in the future.
Step 2: Apply the Quadratic Formula
Now that we have the equation in standard form and have identified the coefficients, we can apply the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
Substitute the values of a, b, and c into the formula:
x = (-1 ± √(1² - 4(1)(-5))) / (2(1))
Now, we simplify the expression step by step. First, simplify the expression inside the square root:
1² - 4(1)(-5) = 1 + 20 = 21
So, the formula becomes:
x = (-1 ± √21) / 2
This step is the heart of solving quadratic equations using the quadratic formula. It requires careful substitution and simplification. The ± sign indicates that there are two possible solutions, one with addition and one with subtraction. It is important to follow the order of operations (PEMDAS/BODMAS) to ensure accurate simplification. Mistakes in this step can lead to incorrect solutions. By practicing this step, you will become more comfortable with the quadratic formula and its application. The ability to confidently apply the quadratic formula is a valuable skill in algebra and will serve you well in future mathematical endeavors.
Step 3: Simplify the Solutions
From the previous step, we have:
x = (-1 ± √21) / 2
This gives us two solutions:
x₁ = (-1 + √21) / 2
x₂ = (-1 - √21) / 2
Since √21 cannot be simplified further (as 21 has no perfect square factors other than 1), these are the simplified solutions. The solutions are irrational numbers because they involve the square root of a non-perfect square. These solutions represent the points where the parabola represented by the quadratic equation intersects the x-axis. Simplifying the solutions is an important step because it presents the answers in their most concise form. It also helps in understanding the nature of the solutions, whether they are rational, irrational, or complex. In this case, the solutions are irrational, indicating that the roots of the quadratic equation are not whole numbers or simple fractions. The ability to simplify solutions is a fundamental skill in mathematics and is essential for solving various types of problems.
The Solutions
The two solutions for the equation x² = 5 - x are:
x₁ = (-1 + √21) / 2
x₂ = (-1 - √21) / 2
These solutions correspond to one of the options provided in the original question.
Conclusion
In this article, we have demonstrated how to use the quadratic formula to solve the quadratic equation x² = 5 - x. We first rewrote the equation in standard form, identified the coefficients, applied the quadratic formula, and then simplified the solutions. The quadratic formula is a powerful tool that can be used to solve any quadratic equation, regardless of its complexity. By understanding the steps involved and practicing regularly, you can master this essential skill and confidently solve quadratic equations. The quadratic formula is not just a mathematical tool; it is a gateway to understanding more advanced concepts in algebra and other areas of mathematics. It is a fundamental building block for solving real-world problems in various fields, such as physics, engineering, and economics. By mastering the quadratic formula, you are equipping yourself with a valuable skill that will serve you well throughout your academic and professional life.
Choosing the Correct Answer
Based on our calculations, the values of x that satisfy the equation x² = 5 - x are:
x₁ = (-1 + √21) / 2
x₂ = (-1 - √21) / 2
These solutions match the option:
(-1 ± √21) / 2
Therefore, this is the correct answer.
