Solving $2x^2 + 7x = -5$ A Comprehensive Guide

Introduction to Quadratic Equations

In the realm of mathematics, quadratic equations hold a significant place, appearing in various contexts ranging from basic algebra to advanced calculus and physics. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The solutions to a quadratic equation, also known as roots or zeros, represent the points where the parabola defined by the equation intersects the x-axis. Understanding how to solve quadratic equations is a fundamental skill in mathematics, paving the way for more complex problem-solving scenarios. There are several methods to tackle these equations, each with its own advantages and suitability depending on the specific equation at hand. These methods include factoring, completing the square, and using the quadratic formula. Factoring involves breaking down the quadratic expression into two linear factors, a method that is particularly effective when the equation can be easily factored. Completing the square is a technique that transforms the equation into a perfect square trinomial, allowing for a straightforward solution by taking the square root. The quadratic formula, a universally applicable method, provides a direct solution using the coefficients of the equation. This guide will delve into the intricacies of solving the quadratic equation $2x^2 + 7x = -5$, demonstrating the application of these methods to arrive at the solution.

Problem Statement: $2x^2 + 7x = -5$

Let's consider the specific quadratic equation we aim to solve: $2x^2 + 7x = -5$. This equation is a classic example of a quadratic equation, exhibiting the characteristic form where the highest power of the variable x is two. However, before we can apply our standard solution methods, we need to manipulate the equation into its general form, which is ax² + bx + c = 0. This form is crucial because it allows us to easily identify the coefficients a, b, and c, which are essential for methods like the quadratic formula and completing the square. By transforming the equation into the general form, we set the stage for a systematic approach to finding the solutions. In our case, the equation $2x^2 + 7x = -5$ needs a slight adjustment to fit this format. The constant term, -5, is on the wrong side of the equation. To rectify this, we need to add 5 to both sides of the equation. This simple algebraic manipulation ensures that we maintain the equality while aligning the equation with the standard quadratic form. This step is not merely cosmetic; it's a fundamental requirement for applying many solution techniques and ensuring accurate results. Once the equation is in the standard form, we can clearly see the values of a, b, and c, allowing us to proceed with the appropriate solution method. In the following sections, we will explore different methods to solve this equation, highlighting the steps involved and the rationale behind each approach. Understanding the initial transformation into the general form is a critical first step in mastering the art of solving quadratic equations.

Method 1: Solving by Factoring

Factoring is a powerful technique for solving quadratic equations, particularly when the equation can be easily expressed as a product of two binomials. This method relies on the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. In the context of quadratic equations, this means that if we can factor the quadratic expression into the form (px + q)(rx + s) = 0, then either px + q = 0 or rx + s = 0, allowing us to solve for x. The key to factoring is to find two numbers that multiply to the constant term (c) and add up to the coefficient of the linear term (b). This process often involves some trial and error, but with practice, it becomes a highly efficient method. For our equation, $2x^2 + 7x + 5 = 0$, we need to find two numbers that multiply to 2 * 5 = 10 and add up to 7. These numbers are 2 and 5. Once we've identified these numbers, we can rewrite the middle term (7x) as the sum of 2x and 5x. This step is crucial because it allows us to group terms and factor by grouping. The rewritten equation becomes $2x^2 + 2x + 5x + 5 = 0$. Now, we can factor out the greatest common factor from the first two terms and the last two terms. From the first two terms, we can factor out 2x, and from the last two terms, we can factor out 5. This gives us 2x(x + 1) + 5(x + 1) = 0$***. Notice that we now have a common factor of ***(x + 1)*** in both terms. Factoring out this common factor, we get ***(2x + 5)(x + 1) = 0$. Now, we can apply the zero-product property. Setting each factor equal to zero, we have 2x + 5 = 0 and x + 1 = 0. Solving these linear equations for x, we find the solutions to the quadratic equation. This method demonstrates the elegance and efficiency of factoring when applicable, providing a clear and direct path to the solutions.

Detailed Steps for Factoring $2x^2 + 7x + 5 = 0$

To effectively solve the quadratic equation $2x^2 + 7x + 5 = 0$ by factoring, we need to break down the process into a series of manageable steps. This methodical approach ensures accuracy and clarity in our solution. The first crucial step is to identify the coefficients a, b, and c in the quadratic equation. In this case, a = 2, b = 7, and c = 5. These coefficients are the foundation for our factoring strategy. Next, we need to find two numbers that satisfy two conditions: their product must equal a * c (which is 2 * 5 = 10), and their sum must equal b (which is 7). This is a critical step, as these numbers will allow us to rewrite the middle term of the quadratic expression. After some consideration, we find that the numbers 2 and 5 meet these criteria. They multiply to 10 and add up to 7. Now, we rewrite the middle term, 7x, as the sum of 2x and 5x. This gives us the equation $2x^2 + 2x + 5x + 5 = 0$. This step is essential for grouping and factoring. The next step is to factor by grouping. We group the first two terms and the last two terms together: (2x^2 + 2x) + (5x + 5) = 0$***. From the first group, we can factor out 2x, and from the second group, we can factor out 5. This results in ***2x(x + 1) + 5(x + 1) = 0$. Notice that we now have a common factor of (x + 1) in both terms. We factor out this common factor, giving us (2x + 5)(x + 1) = 0$***. This is the factored form of the quadratic equation. Finally, we apply the zero-product property. We set each factor equal to zero: ***2x + 5 = 0*** and ***x + 1 = 0$. Solving these linear equations for x, we find the solutions. From 2x + 5 = 0, we subtract 5 from both sides to get 2x = -5$***, and then divide by 2 to get ***x = -5/2$. From x + 1 = 0$***, we subtract 1 from both sides to get ***x = -1$. Therefore, the solutions to the quadratic equation $2x^2 + 7x + 5 = 0$ are x = -5/2 and x = -1$. This detailed step-by-step explanation provides a clear understanding of the factoring method and its application to this specific quadratic equation.

Method 2: Solving Using the Quadratic Formula

The quadratic formula is a universally applicable method for solving quadratic equations of the form ax² + bx + c = 0. Unlike factoring, which relies on the equation being factorable, the quadratic formula can be used to find the solutions for any quadratic equation, regardless of whether it can be factored. This makes it a powerful and reliable tool in algebra. The formula itself is derived from the method of completing the square and is given by: x = [-b ± √(b² - 4ac)] / (2a). Here, a, b, and c are the coefficients of the quadratic equation. The term inside the square root, b² - 4ac, is known as the discriminant. The discriminant provides valuable information about the nature of the solutions. If the discriminant is positive, the equation has two distinct real solutions. If it is zero, the equation has exactly one real solution (a repeated root). If it is negative, the equation has two complex solutions. To apply the quadratic formula, we first need to identify the coefficients a, b, and c from our equation, $2x^2 + 7x + 5 = 0$. In this case, a = 2, b = 7, and c = 5. Once we have identified the coefficients, we substitute these values into the quadratic formula. This involves replacing a, b, and c in the formula with their respective values. After substitution, we simplify the expression, following the order of operations. This typically involves calculating the discriminant first, then performing the addition and subtraction, and finally dividing by 2a. The quadratic formula provides two possible solutions, one obtained by using the plus sign (+) and the other by using the minus sign (-) in front of the square root. These two solutions represent the points where the parabola defined by the quadratic equation intersects the x-axis. The quadratic formula is an indispensable tool in mathematics, offering a systematic and reliable way to solve quadratic equations. Its ability to handle any quadratic equation makes it a fundamental technique for students and professionals alike.

Step-by-Step Application of the Quadratic Formula

To effectively utilize the quadratic formula for solving $2x^2 + 7x + 5 = 0$, a systematic, step-by-step approach is essential. This ensures accuracy and clarity throughout the solution process. Our first critical step involves identifying the coefficients a, b, and c from the quadratic equation. In this specific equation, we have a = 2, b = 7, and c = 5. These coefficients are the foundation upon which we build our solution using the quadratic formula. Next, we substitute these identified values into the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a). Replacing a, b, and c with their respective values, we get x = [-7 ± √(7² - 4 * 2 * 5)] / (2 * 2). This substitution is a crucial step, ensuring that we are working with the correct values in the formula. Now, we simplify the expression following the order of operations. First, we calculate the discriminant, which is the term inside the square root: b² - 4ac = 7² - 4 * 2 * 5 = 49 - 40 = 9$***. The discriminant is 9, which is a positive number, indicating that the equation has two distinct real solutions. Next, we substitute the discriminant back into the formula: ***x = [-7 ± √9] / 4$. Since the square root of 9 is 3, we have x = [-7 ± 3] / 4$***. Now, we calculate the two possible solutions. For the first solution, we use the plus sign: ***x₁ = (-7 + 3) / 4 = -4 / 4 = -1$. For the second solution, we use the minus sign: x₂ = (-7 - 3) / 4 = -10 / 4 = -5/2$. Therefore, the solutions to the quadratic equation $2x^2 + 7x + 5 = 0$ are x = -1 and x = -5/2$. This step-by-step application of the quadratic formula provides a clear and concise method for solving quadratic equations, highlighting the importance of accurate substitution and simplification.

Comparison of Factoring and the Quadratic Formula

Both factoring and the quadratic formula are powerful methods for solving quadratic equations, but they have distinct characteristics that make them suitable for different situations. Factoring is often the quicker method when the quadratic equation can be easily factored. It relies on identifying the factors of the quadratic expression and applying the zero-product property. This method is particularly efficient when the coefficients are small integers and the roots are rational numbers. However, factoring can be challenging or even impossible when the roots are irrational or complex, or when the coefficients are large or not easily factored. On the other hand, the quadratic formula is a universally applicable method that can solve any quadratic equation, regardless of the nature of its roots or the complexity of its coefficients. It provides a direct solution using the coefficients of the equation, making it a reliable tool even when factoring is difficult or impractical. The quadratic formula is especially useful when the roots are irrational or complex, as it provides a straightforward way to calculate these solutions. However, the quadratic formula can be more time-consuming than factoring, especially for simple equations that can be easily factored. It involves more calculations and requires careful attention to detail to avoid errors. In the case of our example equation, $2x^2 + 7x + 5 = 0$, we successfully solved it using both factoring and the quadratic formula. Factoring was relatively straightforward, as the equation could be factored into (2x + 5)(x + 1) = 0$. The quadratic formula also provided the same solutions, but it involved more steps and calculations. Ultimately, the choice between factoring and the quadratic formula depends on the specific equation and the solver's preference. Factoring is often the first method to try for simple equations, while the quadratic formula is a reliable backup for more complex or unfactorable equations. Understanding the strengths and limitations of each method is crucial for efficient problem-solving in mathematics.

Conclusion: Solutions to $2x^2 + 7x = -5$

In conclusion, we have thoroughly explored the process of solving the quadratic equation $2x^2 + 7x = -5$, employing two distinct methods: factoring and the quadratic formula. Through these methods, we have demonstrated that the solutions to the equation are x = -1 and x = -5/2. Factoring provided a direct and efficient approach, leveraging the ability to rewrite the quadratic expression as a product of two binomials. This method highlighted the importance of recognizing factorable patterns and applying the zero-product property to arrive at the solutions. The quadratic formula, on the other hand, offered a universally applicable method, ensuring that we could solve the equation regardless of its factorability. This method underscored the significance of the coefficients in determining the solutions and the power of a formulaic approach in mathematics. By comparing these two methods, we have gained a deeper appreciation for the versatility of quadratic equation solving techniques. Factoring shines when the equation is easily factorable, offering a quicker path to the solutions. The quadratic formula stands as a reliable alternative, capable of handling any quadratic equation, particularly those with irrational or complex roots. The ability to choose the most appropriate method for a given equation is a valuable skill in mathematics, allowing for efficient and effective problem-solving. Mastering these techniques not only equips us to solve quadratic equations but also lays a strong foundation for tackling more advanced mathematical concepts. The solutions we have found, x = -1 and x = -5/2, represent the points where the parabola defined by the equation intersects the x-axis, providing a visual representation of the algebraic solutions. This connection between algebra and geometry is a fundamental aspect of mathematics, enriching our understanding of both disciplines.