Solving 2x^2 + 5x - 15 = 0 A Step-by-Step Guide
In the realm of mathematics, quadratic equations hold a significant position, serving as fundamental tools for modeling and solving a wide array of real-world problems. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to these equations, also known as roots or zeros, represent the values of x that satisfy the equation. Understanding how to solve quadratic equations is crucial for various applications, ranging from physics and engineering to economics and computer science.
Quadratic equations are ubiquitous in various fields due to their ability to model parabolic relationships. For instance, the trajectory of a projectile under the influence of gravity can be described by a quadratic equation. Similarly, the shape of a suspension bridge cable and the curvature of a satellite dish can be modeled using quadratic functions. In economics, quadratic equations can be used to represent cost curves, revenue curves, and profit functions. In computer graphics, quadratic equations play a vital role in creating curves and surfaces.
There are several methods available for solving quadratic equations, each with its own advantages and limitations. The most common methods include factoring, completing the square, and using the quadratic formula. Factoring involves expressing the quadratic expression as a product of two linear factors, which can then be solved individually. Completing the square involves manipulating the equation to create a perfect square trinomial, which can be easily solved by taking the square root. The quadratic formula is a general formula that provides the solutions to any quadratic equation, regardless of whether it can be factored or not. The choice of method often depends on the specific equation and the desired level of accuracy.
In this article, we will delve into the process of solving the specific quadratic equation 2x² + 5x - 15 = 0. This equation serves as an excellent example to illustrate the different techniques used to find the roots of a quadratic equation. We will explore the application of the quadratic formula, which is a universally applicable method for solving quadratic equations. By understanding the steps involved in solving this equation, you will gain valuable insights into the general approach for tackling quadratic equations.
The equation 2x² + 5x - 15 = 0 is a standard quadratic equation in the form ax² + bx + c = 0, where a = 2, b = 5, and c = -15. The coefficients a, b, and c play a crucial role in determining the nature and values of the roots. The coefficient a determines the shape and direction of the parabola represented by the quadratic equation. The coefficient b influences the position of the parabola's axis of symmetry, and the constant term c represents the y-intercept of the parabola. The roots of the quadratic equation are the x-intercepts of the parabola, which are the points where the parabola intersects the x-axis.
Before applying any specific method, it's often helpful to analyze the equation to determine the most efficient approach. In this case, we can attempt to factor the quadratic expression, but it may not be immediately obvious how to do so. Therefore, we can proceed directly to using the quadratic formula, which is a reliable method for solving any quadratic equation. The quadratic formula is derived by completing the square on the general quadratic equation ax² + bx + c = 0. It provides a direct way to calculate the roots without the need for factoring or completing the square.
The quadratic formula is a cornerstone of algebra, providing a universal solution for equations of the form ax² + bx + c = 0. This formula elegantly expresses the roots of the equation in terms of its coefficients, a, b, and c. It's a powerful tool that bypasses the need for factoring or completing the square, especially when dealing with complex or non-factorable quadratic expressions. The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
Where:
- x represents the 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 two possible solutions: one with a plus sign and one with a minus sign.
- The expression inside the square root, b² - 4ac, is called the discriminant, which plays a crucial role in determining the nature of the roots.
To apply the quadratic formula to our equation, 2x² + 5x - 15 = 0, we first identify the coefficients: a = 2, b = 5, and c = -15. We then substitute these values into the quadratic formula:
x = (-5 ± √(5² - 4 * 2 * -15)) / (2 * 2)
Simplifying the expression under the square root:
x = (-5 ± √(25 + 120)) / 4
x = (-5 ± √145) / 4
Now, we have two possible solutions:
x₁ = (-5 + √145) / 4
x₂ = (-5 - √145) / 4
These are the exact solutions to the quadratic equation 2x² + 5x - 15 = 0. They represent the two points where the parabola defined by the equation intersects the x-axis. To obtain approximate numerical values, we can use a calculator to evaluate the square root of 145 and perform the arithmetic operations:
x₁ ≈ (-5 + 12.04) / 4 ≈ 1.76
x₂ ≈ (-5 - 12.04) / 4 ≈ -4.26
Thus, the approximate solutions to the quadratic equation 2x² + 5x - 15 = 0 are x ≈ 1.76 and x ≈ -4.26.
The solutions we obtained, x₁ ≈ 1.76 and x₂ ≈ -4.26, represent the x-intercepts of the parabola defined by the quadratic equation 2x² + 5x - 15 = 0. These are the points where the parabola crosses the x-axis, and they are also known as the roots or zeros of the equation. The solutions provide valuable information about the behavior of the quadratic function and its relationship to the x-axis.
The discriminant, b² - 4ac, is a crucial part of the quadratic formula that provides insight into the nature of the roots. In our case, the discriminant is 145, which is a positive number. A positive discriminant indicates that the quadratic equation has two distinct real roots, which we indeed found. If the discriminant were zero, the equation would have one real root (a repeated root), and if it were negative, the equation would have two complex roots.
The discriminant's value provides a quick way to determine the number and type of solutions without actually solving the equation. For example, if we encounter a quadratic equation with a negative discriminant, we know immediately that the solutions will be complex numbers, and we don't need to proceed with further calculations unless we specifically need the complex solutions.
In the context of the graph of the quadratic equation, the discriminant tells us how many times the parabola intersects the x-axis. A positive discriminant means the parabola intersects the x-axis at two distinct points, a zero discriminant means the parabola touches the x-axis at one point (the vertex), and a negative discriminant means the parabola does not intersect the x-axis at all.
In this comprehensive guide, we have successfully navigated the process of solving the quadratic equation 2x² + 5x - 15 = 0. We employed the quadratic formula, a powerful tool that provides a universal solution for quadratic equations. By carefully substituting the coefficients into the formula and simplifying the expression, we arrived at the exact solutions: x₁ = (-5 + √145) / 4 and x₂ = (-5 - √145) / 4. We also obtained approximate numerical values for these solutions: x ≈ 1.76 and x ≈ -4.26.
Furthermore, we explored the significance of the discriminant, which in this case was 145. The positive discriminant confirmed that the equation has two distinct real roots, corresponding to the two points where the parabola intersects the x-axis. Understanding the discriminant allows us to quickly assess the nature of the solutions without needing to solve the equation completely.
The ability to solve quadratic equations is a fundamental skill in mathematics and has wide-ranging applications in various fields. Whether you're modeling physical phenomena, optimizing economic models, or designing computer graphics, quadratic equations often play a crucial role. By mastering the quadratic formula and understanding the discriminant, you'll be well-equipped to tackle a wide range of mathematical problems and real-world applications.
This exploration has provided a solid foundation for understanding and solving quadratic equations. With practice and further exploration, you can deepen your understanding and expand your mathematical toolkit.
