Solving 4x² + X - 3 = 0 Using The Quadratic Formula A Step-by-Step Guide

Introduction: Understanding Quadratic Equations

In the realm of mathematics, quadratic equations hold a significant place. These equations, characterized by their highest power being a square (x²), frequently arise in various scientific and engineering applications. Understanding how to solve quadratic equations is therefore a fundamental skill. One of the most reliable and universally applicable methods for solving them is the quadratic formula. This article will delve into the process of using the quadratic formula to solve the specific equation 4x² + x - 3 = 0, providing a comprehensive, step-by-step guide suitable for anyone looking to master this crucial algebraic technique. Before diving into the solution, let's first establish a firm understanding of what a quadratic equation is and why the quadratic formula is such a powerful tool in our mathematical arsenal.

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 we aim to solve for. The coefficient 'a' is particularly important, as it cannot be zero; otherwise, the equation would reduce to a linear equation. The presence of the x² term is what distinguishes a quadratic equation from other types of algebraic equations.

There are several methods for solving quadratic equations, including factoring, completing the square, and the quadratic formula. Factoring, when applicable, is often the quickest method, but it's not always feasible, especially when the roots are not rational numbers. Completing the square is a more versatile method, but it can be cumbersome and prone to errors if not executed carefully. The quadratic formula, on the other hand, provides a direct and reliable solution for any quadratic equation, regardless of the nature of its roots. This makes it an indispensable tool in mathematics.

The quadratic formula is derived from the method of completing the square and offers a straightforward way to find the roots (or solutions) of any quadratic equation. It elegantly expresses the solutions in terms of the coefficients 'a', 'b', and 'c'. The formula itself is a compact and powerful expression, and understanding its derivation can provide deeper insights into its application. In the following sections, we will explore the quadratic formula in detail and demonstrate its application to solving the equation 4x² + x - 3 = 0.

By the end of this article, you will not only be able to solve this specific quadratic equation but also gain a thorough understanding of the quadratic formula and its applications. This knowledge will empower you to tackle a wide range of quadratic equations with confidence and precision. Whether you are a student learning algebra for the first time or someone looking to refresh your mathematical skills, this guide will provide you with the tools and understanding you need to succeed.

The Quadratic Formula: Unveiling the Solution

The quadratic formula is a cornerstone of algebra, providing a direct method for solving quadratic equations of the form ax² + bx + c = 0. It is expressed as:

x = (-b ± √(b² - 4ac)) / 2a

This formula might seem daunting at first glance, but its power lies in its ability to provide the solutions (also known as roots or zeros) of any quadratic equation, regardless of whether the solutions are real or complex, rational or irrational. The ± symbol indicates that there are typically two solutions, one obtained by adding the square root term and the other by subtracting it. Understanding each component of the formula is crucial for its correct application.

The coefficients a, b, and c are the numerical values that define the quadratic equation. 'a' is the coefficient of the x² term, 'b' is the coefficient of the x term, and 'c' is the constant term. These coefficients are the key inputs to the quadratic formula, and correctly identifying them is the first step in solving any quadratic equation using this method. A common mistake is to overlook the signs of these coefficients, which can lead to incorrect solutions. For instance, if the equation is 2x² - 3x + 1 = 0, then a = 2, b = -3, and c = 1. Note the negative sign associated with b.

The expression inside the square root, b² - 4ac, is known as the discriminant. The discriminant plays a crucial role in determining the nature of the roots of the quadratic equation. It provides valuable information about the number and type of solutions without actually solving the equation. There are three possible scenarios:

1. If b² - 4ac > 0, the equation has two distinct real roots. This means the quadratic equation intersects the x-axis at two different points.
2. If b² - 4ac = 0, the equation has exactly one real root (a repeated root). In this case, the quadratic equation touches the x-axis at exactly one point.
3. If b² - 4ac < 0, the equation has two complex roots. This indicates that the quadratic equation does not intersect the x-axis.

Understanding the discriminant allows us to predict the type of solutions we will obtain before even applying the full quadratic formula. This can be a valuable check on our work, ensuring that the solutions we calculate are consistent with the nature of the equation.

The denominator of the quadratic formula, 2a, is simply twice the coefficient of the x² term. This term ensures that the solutions are properly scaled and accounts for the quadratic nature of the equation. It is important to remember that 'a' cannot be zero, as this would result in division by zero, rendering the formula undefined. This is consistent with the definition of a quadratic equation, where 'a' must be non-zero.

In the next section, we will apply the quadratic formula to the specific equation 4x² + x - 3 = 0. We will carefully identify the coefficients, substitute them into the formula, and simplify the expression to find the solutions. By following this step-by-step process, you will gain a clear understanding of how to use the quadratic formula effectively.

Solving 4x² + x - 3 = 0: A Step-by-Step Application

Now, let's put the quadratic formula into action by solving the equation 4x² + x - 3 = 0. This section will provide a detailed, step-by-step application of the formula, ensuring clarity and understanding at each stage. The key to success with the quadratic formula lies in careful substitution and simplification.

The first step is to identify the coefficients a, b, and c in the given equation. Comparing 4x² + x - 3 = 0 with the general form ax² + bx + c = 0, we can see that:

• a = 4
• b = 1
• c = -3

It is crucial to pay close attention to the signs of the coefficients. In this case, 'c' is negative, which will affect the subsequent calculations. A common mistake is to overlook the negative sign, leading to incorrect solutions. Double-checking these values before proceeding is always a good practice.

Next, we substitute these values into the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

x = (-1 ± √(1² - 4 * 4 * -3)) / (2 * 4)

This step involves replacing the variables a, b, and c in the formula with their corresponding numerical values. Parentheses are used to ensure that the operations are performed in the correct order, especially when dealing with negative numbers. This substitution is the most critical step, as any error here will propagate through the rest of the solution.

Now, we simplify the expression step by step. First, let's simplify the expression inside the square root:

1² - 4 * 4 * -3 = 1 + 48 = 49

The square root of 49 is 7, which simplifies the formula further. The discriminant, in this case, is positive (49 > 0), indicating that the equation has two distinct real roots. This aligns with our understanding of the discriminant's role in determining the nature of the roots.

Now, substitute the simplified value back into the quadratic formula:

x = (-1 ± √49) / 8

x = (-1 ± 7) / 8

This gives us two possible solutions, one with the plus sign and one with the minus sign. Let's calculate each solution separately:

For the plus sign:

x = (-1 + 7) / 8 = 6 / 8 = 3 / 4

For the minus sign:

x = (-1 - 7) / 8 = -8 / 8 = -1

Therefore, the solutions to the quadratic equation 4x² + x - 3 = 0 are x = 3/4 and x = -1. These are the two values of x that satisfy the equation, meaning that if we substitute either of these values back into the original equation, the equation will hold true.

In conclusion, by carefully identifying the coefficients, substituting them into the quadratic formula, and simplifying the resulting expression, we have successfully solved the equation 4x² + x - 3 = 0. This step-by-step approach highlights the power and versatility of the quadratic formula in solving quadratic equations.

Verifying the Solutions and Their Significance

After solving a quadratic equation, it's crucial to verify the solutions to ensure their correctness. This step not only confirms that the calculations were accurate but also deepens our understanding of what the solutions represent. Verifying solutions involves substituting each solution back into the original equation and checking if the equation holds true. This process is a fundamental aspect of problem-solving in mathematics and helps prevent errors.

Let's verify the solutions x = 3/4 and x = -1 for the equation 4x² + x - 3 = 0. First, we'll substitute x = 3/4 into the equation:

4(3/4)² + (3/4) - 3 = 0

4(9/16) + (3/4) - 3 = 0

9/4 + 3/4 - 3 = 0

12/4 - 3 = 0

3 - 3 = 0

0 = 0

Since the equation holds true, x = 3/4 is a valid solution. Now, let's verify x = -1:

4(-1)² + (-1) - 3 = 0

4(1) - 1 - 3 = 0

4 - 1 - 3 = 0

0 = 0

Again, the equation holds true, confirming that x = -1 is also a valid solution. This verification process assures us that our application of the quadratic formula and the subsequent simplifications were accurate.

The solutions to a quadratic equation have significant interpretations in various contexts. Geometrically, the solutions represent the x-intercepts (or roots) of the parabola defined by the quadratic equation. In other words, they are the points where the graph of the equation y = 4x² + x - 3 intersects the x-axis. These points are critical in understanding the behavior of the quadratic function.

In practical applications, the solutions can represent a variety of real-world phenomena. For example, in physics, quadratic equations are often used to model projectile motion, and the solutions can represent the time it takes for a projectile to reach a certain height or the distance it travels. In engineering, quadratic equations can arise in circuit analysis, structural design, and other fields, where the solutions may represent critical parameters or design constraints.

Understanding the significance of the solutions goes beyond simply finding the numerical values. It involves interpreting them in the context of the problem and recognizing their implications. For instance, if a quadratic equation is used to model the profit of a business, the solutions might represent the break-even points, where the profit is zero. Analyzing these points can provide valuable insights for decision-making.

In summary, verifying the solutions to a quadratic equation is a crucial step in the problem-solving process. It not only confirms the accuracy of the calculations but also reinforces our understanding of the solutions' meaning and significance. The solutions themselves can have important geometric interpretations and practical applications in various fields, highlighting the importance of mastering quadratic equations and their solutions.

Conclusion: Mastering Quadratic Equations

In conclusion, this article has provided a comprehensive guide to solving quadratic equations using the quadratic formula, with a specific focus on the equation 4x² + x - 3 = 0. We have explored the fundamental concepts of quadratic equations, delved into the quadratic formula itself, applied it step-by-step to find the solutions, and verified these solutions to ensure their accuracy. Mastering quadratic equations is an essential skill in mathematics, and the quadratic formula stands as a powerful tool in this endeavor.

We began by defining quadratic equations and highlighting their significance in various fields. Understanding the general form ax² + bx + c = 0 and the role of the coefficients a, b, and c is the foundation for solving these equations. We discussed different methods for solving quadratic equations, including factoring, completing the square, and the quadratic formula, emphasizing the versatility and reliability of the quadratic formula.

The heart of the article was the detailed explanation and application of the quadratic formula. We carefully dissected the formula, highlighting the importance of each component, including the discriminant. The discriminant, b² - 4ac, provides valuable information about the nature of the roots, allowing us to predict whether the equation has two distinct real roots, one repeated real root, or two complex roots. This understanding enhances our ability to solve and interpret quadratic equations effectively.

Applying the quadratic formula to the equation 4x² + x - 3 = 0, we demonstrated a step-by-step approach to finding the solutions. This process involved identifying the coefficients, substituting them into the formula, simplifying the expression, and ultimately arriving at the solutions x = 3/4 and x = -1. This detailed walkthrough provides a clear model for solving other quadratic equations using the same method.

Verification of the solutions is a critical step in the problem-solving process. By substituting the solutions back into the original equation, we confirmed their correctness and reinforced our understanding of what it means for a value to be a solution. This practice instills confidence in our results and helps prevent errors.

Beyond the mechanics of solving quadratic equations, we explored the significance of the solutions. We discussed their geometric interpretation as x-intercepts of a parabola and their practical applications in various fields, such as physics and engineering. Understanding the context and implications of the solutions adds depth to our mathematical understanding and allows us to apply these concepts in real-world scenarios.

In essence, the quadratic formula is a powerful tool that empowers us to solve a wide range of quadratic equations. By mastering this formula and the underlying concepts, we gain a valuable skill that is applicable in various areas of mathematics and beyond. Whether you are a student learning algebra or someone looking to refresh your mathematical skills, the knowledge and techniques presented in this article will serve as a solid foundation for your continued mathematical journey. The ability to confidently solve quadratic equations opens doors to more advanced mathematical concepts and problem-solving opportunities, making it a truly worthwhile skill to acquire.