Solving X^2 + X - 10 = 0 Using The Quadratic Formula A Step By Step Guide

The quadratic formula is a powerful tool for solving quadratic equations of the form ax² + bx + c = 0. In this comprehensive guide, we will delve into the intricacies of using the quadratic formula to solve equations, providing a step-by-step approach and addressing common challenges. We will specifically tackle the equation x² + x - 10 = 0, demonstrating the application of the formula and the interpretation of the results. Understanding the quadratic formula is crucial for various mathematical and scientific applications, including physics, engineering, and economics. It allows us to find the roots, or solutions, of quadratic equations, which represent the points where the parabola intersects the x-axis. The quadratic formula is not just a mathematical equation; it's a cornerstone of problem-solving in numerous fields. Let's embark on this journey to master the quadratic formula and unlock its potential.

Understanding Quadratic Equations and the Quadratic Formula

Before diving into the solution, let's establish a solid understanding of quadratic equations and the quadratic formula itself. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. These constants are the coefficients of the quadratic equation, and they play a crucial role in determining the nature and values of the solutions. The quadratic formula is derived from the method of completing the square and provides a direct way to find the solutions for any quadratic equation. The formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

In this formula:

• x represents the solutions or roots of the quadratic equation.
• a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.
• The ± symbol indicates that there are potentially two solutions: one obtained by adding the square root term and the other by subtracting it.
• The expression b² - 4ac under the square root is called the discriminant. The discriminant provides valuable information about the nature of the solutions: a positive discriminant indicates two distinct real solutions, a zero discriminant indicates one real solution (a repeated root), and a negative discriminant indicates two complex solutions.

The quadratic formula is a versatile tool that can be applied to any quadratic equation, regardless of the complexity of its coefficients. It eliminates the need for factoring or completing the square, making it a reliable and efficient method for finding solutions. Now that we have a firm grasp of the quadratic formula, let's apply it to the given equation.

Applying the Quadratic Formula to Solve x² + x - 10 = 0

Our goal is to solve the quadratic equation x² + x - 10 = 0 using the quadratic formula. To do this, we first need to identify the coefficients a, b, and c. By comparing the given equation to the general form ax² + bx + c = 0, we can see that:

• a = 1 (the coefficient of x²)
• b = 1 (the coefficient of x)
• c = -10 (the constant term)

Now that we have identified the coefficients, we can substitute them into the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

Substituting the values, we get:

x = (-1 ± √(1² - 4 * 1 * -10)) / (2 * 1)

Next, we simplify the expression inside the square root:

x = (-1 ± √(1 + 40)) / 2

x = (-1 ± √41) / 2

Now we have two possible solutions:

• x₁ = (-1 + √41) / 2
• x₂ = (-1 - √41) / 2

These are the two roots of the quadratic equation x² + x - 10 = 0. They represent the points where the parabola defined by the equation intersects the x-axis. We can approximate these solutions to decimal values for practical applications or for graphing purposes. The solutions obtained using the quadratic formula are exact, and they provide a complete understanding of the equation's behavior. In the next section, we will explore how to interpret these solutions and what they tell us about the quadratic equation.

Interpreting the Solutions

The solutions we obtained, x₁ = (-1 + √41) / 2 and x₂ = (-1 - √41) / 2, are the roots of the quadratic equation x² + x - 10 = 0. These roots represent the x-intercepts of the parabola defined by the equation y = x² + x - 10. In other words, they are the points where the parabola crosses the x-axis. The fact that we obtained two distinct real solutions indicates that the parabola intersects the x-axis at two different points. If the discriminant (the value under the square root, b² - 4ac) were zero, we would have obtained one real solution, meaning the parabola touches the x-axis at only one point (the vertex). If the discriminant were negative, we would have obtained two complex solutions, indicating that the parabola does not intersect the x-axis at any real points.

To get a better understanding of the solutions, we can approximate them to decimal values:

• x₁ = (-1 + √41) / 2 ≈ (-1 + 6.403) / 2 ≈ 2.701
• x₂ = (-1 - √41) / 2 ≈ (-1 - 6.403) / 2 ≈ -3.701

These decimal approximations allow us to visualize the solutions on a number line or a graph. We can see that one solution is positive, and the other is negative. This information can be useful in various applications, such as determining the feasible range of values in a physical problem. The solutions to a quadratic equation are not just abstract numbers; they have real-world significance. They can represent physical quantities, such as time, distance, or area, and they can be used to model and solve a wide range of problems. Understanding how to interpret the solutions is as important as knowing how to find them. In the next section, we will discuss some common mistakes to avoid when using the quadratic formula.

Common Mistakes to Avoid

While the quadratic formula is a powerful tool, it's essential to use it correctly to avoid common mistakes. Here are some pitfalls to watch out for:

1. Incorrectly Identifying Coefficients: The first step in using the quadratic formula is to identify the coefficients a, b, and c. A common mistake is to misidentify these coefficients, especially when the equation is not in the standard form ax² + bx + c = 0. For example, if the equation is given as 10 = x² + x, it's crucial to rearrange it to x² + x - 10 = 0 before identifying a = 1, b = 1, and c = -10. Always ensure the equation is in standard form before extracting the coefficients.
2. Sign Errors: Sign errors are a frequent cause of mistakes when using the quadratic formula. Pay close attention to the signs of the coefficients and the operations within the formula. For instance, the formula includes -b, so if b is negative, -b will be positive. Similarly, the term -4ac can be tricky if either a or c is negative. Double-check all signs to ensure accuracy.
3. Incorrectly Evaluating the Discriminant: The discriminant, b² - 4ac, is a crucial part of the quadratic formula, and errors in its evaluation can lead to incorrect solutions. Make sure to follow the order of operations (PEMDAS/BODMAS) when calculating the discriminant. First, square b, then multiply 4, a, and c, and finally subtract the product from b². A common mistake is to forget the negative sign in -4ac when a or c is negative.
4. Forgetting the ± Sign: The ± sign in the quadratic formula indicates that there are potentially two solutions. Forgetting this sign will result in only one solution being found, which is incorrect if the equation has two distinct roots. Remember to consider both the positive and negative cases when evaluating the formula.
5. Simplification Errors: After applying the quadratic formula, it's essential to simplify the solutions as much as possible. Errors in simplification can occur when dealing with square roots, fractions, or negative signs. Double-check each step of the simplification process to ensure accuracy. For example, if you have a square root that can be simplified (e.g., √4), do so before proceeding with further calculations.

By being aware of these common mistakes and taking steps to avoid them, you can increase your accuracy and confidence in using the quadratic formula. Always double-check your work and, if possible, verify your solutions by substituting them back into the original equation.

Conclusion

The quadratic formula is an indispensable tool for solving quadratic equations. By understanding the formula, its components, and the potential pitfalls, you can confidently tackle a wide range of quadratic equations. In this guide, we have walked through the process of solving the equation x² + x - 10 = 0 using the quadratic formula, highlighting each step and emphasizing the importance of accuracy. Remember to identify the coefficients correctly, pay attention to signs, evaluate the discriminant carefully, consider both solutions from the ± sign, and simplify your answers thoroughly. With practice and attention to detail, you can master the quadratic formula and unlock its power to solve mathematical problems.

This method not only provides the solutions but also offers insights into the nature of the roots and the behavior of the quadratic function. Whether you are a student learning algebra or a professional applying mathematical principles, the quadratic formula is a valuable asset in your problem-solving toolkit. Embrace the formula, practice its application, and you will find yourself equipped to handle quadratic equations with ease and precision.