Solving X² + 2x - 15 = 0 A Comprehensive Guide
In the realm of mathematics, quadratic equations hold a position of paramount importance. These equations, characterized by their second-degree polynomial form, frequently arise in diverse fields, ranging from physics and engineering to economics and computer science. Mastering the techniques for solving quadratic equations is, therefore, an indispensable skill for anyone venturing into these disciplines. This article delves into a comprehensive exploration of how to solve the specific quadratic equation x² + 2x - 15 = 0, elucidating the underlying concepts and providing step-by-step guidance.
Understanding Quadratic Equations
Before we embark on the solution process, it's crucial to establish a solid understanding of what quadratic equations are and the various methods available for solving them. A quadratic equation is an equation 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 roots correspond to the points where the graph of the quadratic function (y = ax² + bx + c) intersects the x-axis.
There are several methods for solving quadratic equations, each with its own strengths and weaknesses. The most commonly used techniques include:
- Factoring: This method involves expressing the quadratic expression as a product of two linear factors. It is an efficient technique when the factors are easily discernible.
- Completing the square: This method transforms the quadratic equation into a perfect square trinomial, allowing for direct solution by taking the square root of both sides.
- Quadratic formula: This formula provides a general solution for any quadratic equation, regardless of whether it can be factored or not. It is a reliable and versatile method.
Solving x² + 2x - 15 = 0 by Factoring
Let's begin by attempting to solve the given equation, x² + 2x - 15 = 0, using the factoring method. This technique hinges on the ability to decompose the quadratic expression into two linear factors. The key idea is to find two numbers that, when multiplied, yield the constant term (-15) and, when added, yield the coefficient of the linear term (2).
In this case, the numbers 5 and -3 satisfy these conditions:
- 5 × (-3) = -15
- 5 + (-3) = 2
Therefore, we can rewrite the quadratic expression as follows:
x² + 2x - 15 = (x + 5)(x - 3)
Now, the equation becomes:
(x + 5)(x - 3) = 0
For the product of two factors to be zero, at least one of the factors must be zero. This leads to two possible scenarios:
- x + 5 = 0, which implies x = -5
- x - 3 = 0, which implies x = 3
Thus, the solutions to the equation x² + 2x - 15 = 0 are x = -5 and x = 3. These values represent the roots of the quadratic equation and the x-intercepts of the corresponding parabola.
Solving x² + 2x - 15 = 0 by Completing the Square
Another powerful method for solving quadratic equations is completing the square. This technique involves manipulating the equation to create a perfect square trinomial on one side, which can then be easily solved by taking the square root. Let's apply this method to our equation, x² + 2x - 15 = 0.
The first step is to move the constant term to the right side of the equation:
x² + 2x = 15
Next, we need to complete the square on the left side. To do this, we take half of the coefficient of the x term (which is 2), square it (1² = 1), and add it to both sides of the equation:
x² + 2x + 1 = 15 + 1
The left side now represents a perfect square trinomial, which can be factored as (x + 1)²:
(x + 1)² = 16
Taking the square root of both sides, we get:
x + 1 = ±4
This gives us two possible equations:
- x + 1 = 4, which implies x = 3
- x + 1 = -4, which implies x = -5
As we found using factoring, the solutions to the equation x² + 2x - 15 = 0 are x = -5 and x = 3. Completing the square provides an alternative approach to solving quadratic equations, particularly when factoring is not straightforward.
Solving x² + 2x - 15 = 0 Using the Quadratic Formula
The quadratic formula is a universal tool for solving quadratic equations, regardless of their factorability. It provides a direct solution for any equation in the form ax² + bx + c = 0. The formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
In our case, the equation is x² + 2x - 15 = 0, so a = 1, b = 2, and c = -15. Plugging these values into the quadratic formula, we get:
x = (-2 ± √(2² - 4 * 1 * -15)) / (2 * 1)
Simplifying the expression under the square root:
x = (-2 ± √(4 + 60)) / 2
x = (-2 ± √64) / 2
Since √64 = 8, we have:
x = (-2 ± 8) / 2
This gives us two possible solutions:
- x = (-2 + 8) / 2 = 6 / 2 = 3
- x = (-2 - 8) / 2 = -10 / 2 = -5
Once again, we arrive at the same solutions: x = -5 and x = 3. The quadratic formula serves as a powerful and reliable method for solving quadratic equations, especially when other techniques prove challenging.
Verifying the Solutions
To ensure the accuracy of our solutions, it's always a good practice to verify them by substituting them back into the original equation. Let's check if x = -5 and x = 3 satisfy the equation x² + 2x - 15 = 0.
For x = -5:
(-5)² + 2(-5) - 15 = 25 - 10 - 15 = 0
For x = 3:
(3)² + 2(3) - 15 = 9 + 6 - 15 = 0
Both solutions satisfy the equation, confirming their validity. This step reinforces our confidence in the accuracy of our calculations and the correctness of the obtained roots.
Conclusion
In this comprehensive guide, we have explored three distinct methods for solving the quadratic equation x² + 2x - 15 = 0: factoring, completing the square, and the quadratic formula. Each method offers a unique approach to finding the roots of the equation, and mastering all three techniques enhances your problem-solving capabilities in mathematics and related fields. We have demonstrated that the solutions to the equation are x = -5 and x = 3, and we have verified these solutions by substituting them back into the original equation.
Understanding quadratic equations and their solutions is a fundamental concept in mathematics. By mastering the techniques presented in this article, you will be well-equipped to tackle a wide range of problems involving quadratic equations and their applications. Remember to practice these methods regularly to solidify your understanding and enhance your problem-solving skills. The journey of mathematical exploration is a continuous one, and quadratic equations serve as a stepping stone towards more advanced concepts and applications.
Keywords: quadratic equations, factoring, completing the square, quadratic formula, roots, solutions, mathematics