Solution Set Of 2x² - 3x = 5 Explained
Introduction
In mathematics, solving equations is a fundamental skill. Among various types of equations, quadratic equations hold a significant place due to their frequent appearance in diverse mathematical and real-world problems. This article delves into the process of finding the solution set of the quadratic equation 2x² - 3x = 5. We will explore different methods to solve this equation, including factoring, completing the square, and using the quadratic formula. Understanding these methods is crucial for anyone studying algebra and related fields. By the end of this discussion, you will have a clear understanding of how to approach and solve quadratic equations, along with a comprehensive understanding of the solution set for the given equation.
Understanding Quadratic Equations
Before diving into the specific equation, let's establish a foundational understanding of quadratic equations. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is expressed as ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' represents the variable. The term 'a' cannot be zero, as that would reduce the equation to a linear form. The solutions to a quadratic equation, also known as roots or zeros, are the values of 'x' that satisfy the equation. These solutions represent the points where the parabola, defined by the quadratic expression, intersects the x-axis. A quadratic equation can have two real solutions, one real solution (a repeated root), or two complex solutions. The nature of the solutions depends on the discriminant, which is given by the formula Δ = b² - 4ac. If Δ > 0, there are two distinct real solutions; if Δ = 0, there is one real solution; and if Δ < 0, there are two complex solutions. In the context of problem-solving, identifying the type of solutions expected can help streamline the solving process. The methods used to find these solutions vary, and understanding each method's strengths and weaknesses is key to efficient problem-solving. This article will cover these methods in detail, ensuring you have a comprehensive toolkit for tackling quadratic equations.
Transforming the Equation
To begin solving the given equation, 2x² - 3x = 5, the first step is to transform it into the standard quadratic form, which is ax² + bx + c = 0. This transformation involves moving all terms to one side of the equation, leaving zero on the other side. By subtracting 5 from both sides of the equation, we get: 2x² - 3x - 5 = 0. Now, the equation is in the standard form, where a = 2, b = -3, and c = -5. This form is essential for applying various methods of solving quadratic equations, such as factoring, completing the square, and using the quadratic formula. Recognizing the coefficients a, b, and c is crucial for each method. For instance, in factoring, these coefficients help in identifying the correct factors. In the quadratic formula, they are directly substituted into the formula to find the solutions. This initial transformation is a foundational step that simplifies the subsequent solving process. It ensures that the equation is in a format that is conducive to the application of established methods. Moreover, it helps in visualizing the equation as a parabola, which can provide additional insights into the nature of the solutions. By mastering this transformation, you lay a strong groundwork for solving a wide range of quadratic equations.
Method 1: Factoring
One of the most efficient methods for solving quadratic equations is factoring, provided the equation can be factored easily. Factoring involves expressing the quadratic expression as a product of two linear expressions. To factor the equation 2x² - 3x - 5 = 0, we look for two binomials (px + q) and (rx + s) such that their product equals the quadratic expression. The process typically involves finding two numbers that multiply to give the product of the leading coefficient (a) and the constant term (c), and add up to the middle coefficient (b). In this case, we need two numbers that multiply to (2)(-5) = -10 and add up to -3. These numbers are -5 and 2. Using these numbers, we can rewrite the middle term of the quadratic expression as -5x + 2x. The equation then becomes 2x² - 5x + 2x - 5 = 0. Next, we factor by grouping: x(2x - 5) + 1(2x - 5) = 0. This simplifies to (x + 1)(2x - 5) = 0. Setting each factor equal to zero gives us two linear equations: x + 1 = 0 and 2x - 5 = 0. Solving these equations, we find the solutions x = -1 and x = 5/2. Thus, factoring allows us to directly find the roots of the equation by decomposing it into simpler, solvable parts. This method is particularly effective when the roots are rational numbers, making it a valuable tool in solving quadratic equations.
Method 2: Quadratic Formula
When factoring is not straightforward or possible, the quadratic formula provides a reliable method for solving quadratic equations. The quadratic formula is derived from the method of completing the square and can be applied to any quadratic equation in the standard form ax² + bx + c = 0. The formula is given by: x = (-b ± √(b² - 4ac)) / (2a). For our equation, 2x² - 3x - 5 = 0, we have a = 2, b = -3, and c = -5. Substituting these values into the formula, we get: x = (-(-3) ± √((-3)² - 4(2)(-5))) / (2(2)). Simplifying, we have x = (3 ± √(9 + 40)) / 4, which further simplifies to x = (3 ± √49) / 4. The square root of 49 is 7, so the solutions are x = (3 ± 7) / 4. This gives us two possible solutions: x = (3 + 7) / 4 = 10 / 4 = 5/2 and x = (3 - 7) / 4 = -4 / 4 = -1. The quadratic formula is particularly useful when the roots are irrational or complex, as it provides a direct way to calculate the solutions without the need for factoring. It is a powerful tool that guarantees a solution for any quadratic equation, making it an essential method in solving mathematical problems.
Determining the Solution Set
After employing the factoring method and the quadratic formula, we have arrived at the solutions for the equation 2x² - 3x = 5. Both methods yielded the same solutions: x = -1 and x = 5/2. The solution set is the set of all values that satisfy the equation. In this case, the solution set consists of two distinct real numbers. We can express the solution set using set notation as -1, 5/2}. This set represents the two points where the parabola defined by the equation 2x² - 3x - 5 intersects the x-axis. Graphically, these points are the x-intercepts of the parabola. To verify that these solutions are correct, we can substitute each value back into the original equation and check if the equation holds true. For x = -1 accurately represents the values of x that make the equation 2x² - 3x = 5 true. Understanding how to determine and express the solution set is a fundamental skill in algebra, providing a concise way to represent the answers to an equation.
Conclusion
In conclusion, we have successfully found the solution set of the quadratic equation 2x² - 3x = 5. By transforming the equation into standard form and applying two different methods – factoring and the quadratic formula – we determined that the solutions are x = -1 and x = 5/2. The solution set, {-1, 5/2}, represents the values that satisfy the equation. This process highlights the importance of understanding various methods for solving quadratic equations, as different methods may be more efficient depending on the equation's characteristics. Factoring is a straightforward method when applicable, while the quadratic formula provides a universal approach for solving any quadratic equation. The ability to transform an equation into standard form is a crucial initial step, enabling the application of these solution methods. Moreover, verifying the solutions by substituting them back into the original equation ensures accuracy and reinforces understanding. Mastering these techniques is essential for success in algebra and related fields, as quadratic equations are a fundamental concept with wide-ranging applications. This exploration not only provides a solution to the specific equation but also enhances your problem-solving skills and understanding of quadratic equations in general.
