Solving 2t^2 + 2t - 3 = 0 A Step-by-Step Guide

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Introduction

In this article, we will delve into the process of finding the solutions to the quadratic equation 2t^2 + 2t - 3 = 0. Quadratic equations, which are polynomial equations of the second degree, play a crucial role in various fields of mathematics, physics, engineering, and computer science. Understanding how to solve them is therefore essential. We will explore different methods to find the solutions, with a particular focus on using the quadratic formula. We will also verify each step meticulously to ensure accuracy and clarity. This comprehensive guide aims to provide you with a clear understanding of the solution process, enabling you to confidently tackle similar problems in the future. Furthermore, we will discuss the nature of the solutions and what they represent graphically, offering a holistic view of quadratic equations.

Understanding the Quadratic Equation

To begin, it's crucial to understand the standard form of a quadratic equation, which is at^2 + bt + c = 0, where a, b, and c are constants, and 't' is the variable. In our case, the equation 2t^2 + 2t - 3 = 0 aligns perfectly with this form, where a = 2, b = 2, and c = -3. The solutions to a quadratic equation, also known as roots or zeros, represent the values of 't' that satisfy the equation. These solutions can be found using several methods, including factoring, completing the square, and the quadratic formula. However, factoring is not always straightforward, especially when dealing with non-integer roots. Completing the square is a powerful method but can be time-consuming. Therefore, the quadratic formula often provides the most efficient route to finding the solutions. It's also important to recognize that a quadratic equation can have two distinct real solutions, one repeated real solution, or two complex solutions, depending on the discriminant (b^2 - 4ac). This characteristic makes understanding the solutions even more crucial.

Applying the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form at^2 + bt + c = 0. This formula is given by: t = (-b ± √(b^2 - 4ac)) / (2a). In our equation, 2t^2 + 2t - 3 = 0, we have a = 2, b = 2, and c = -3. Substituting these values into the quadratic formula, we get: t = (-2 ± √(2^2 - 4 * 2 * (-3))) / (2 * 2). The next step involves simplifying the expression under the square root: 2^2 - 4 * 2 * (-3) = 4 + 24 = 28. So, the formula becomes: t = (-2 ± √28) / 4. We can further simplify √28 by factoring out the largest perfect square, which is 4. Thus, √28 = √(4 * 7) = 2√7. Now, our equation looks like this: t = (-2 ± 2√7) / 4. Finally, we can divide both the numerator and the denominator by 2 to simplify the expression, which gives us: t = (-1 ± √7) / 2. This simplified form directly gives us the two solutions to the equation.

Step-by-Step Solution

Let's break down the solution process step-by-step to ensure clarity:

  1. Identify the coefficients: In the equation 2t^2 + 2t - 3 = 0, we have a = 2, b = 2, and c = -3.
  2. Apply the quadratic formula: The quadratic formula is t = (-b ± √(b^2 - 4ac)) / (2a). Substituting the values, we get t = (-2 ± √(2^2 - 4 * 2 * (-3))) / (2 * 2).
  3. Simplify the discriminant: Calculate the value inside the square root: 2^2 - 4 * 2 * (-3) = 4 + 24 = 28. The equation becomes t = (-2 ± √28) / 4.
  4. Simplify the square root: Simplify √28 as √(4 * 7) = 2√7. The equation is now t = (-2 ± 2√7) / 4.
  5. Reduce the fraction: Divide both the numerator and the denominator by 2: t = (-1 ± √7) / 2.
  6. Identify the solutions: The two solutions are t = (-1 + √7) / 2 and t = (-1 - √7) / 2.

This detailed step-by-step approach ensures that each part of the solution is clear and easy to follow. By breaking down the problem into smaller, manageable steps, we minimize the chances of error and enhance understanding.

Verification of the Solution

To ensure the correctness of our solutions, we can substitute them back into the original equation, 2t^2 + 2t - 3 = 0. Let's take the first solution, t = (-1 + √7) / 2:

  1. Substitute into the equation: 2((-1 + √7) / 2)^2 + 2((-1 + √7) / 2) - 3
  2. Simplify the squared term: 2((1 - 2√7 + 7) / 4) + (-1 + √7) - 3
  3. Further simplification: (8 - 2√7) / 2 + (-1 + √7) - 3
  4. Divide by 2: 4 - √7 - 1 + √7 - 3
  5. Combine like terms: (4 - 1 - 3) + (-√7 + √7) = 0

Now let's verify the second solution, t = (-1 - √7) / 2:

  1. Substitute into the equation: 2((-1 - √7) / 2)^2 + 2((-1 - √7) / 2) - 3
  2. Simplify the squared term: 2((1 + 2√7 + 7) / 4) + (-1 - √7) - 3
  3. Further simplification: (8 + 2√7) / 2 + (-1 - √7) - 3
  4. Divide by 2: 4 + √7 - 1 - √7 - 3
  5. Combine like terms: (4 - 1 - 3) + (√7 - √7) = 0

Both solutions satisfy the original equation, confirming that our solutions are correct. This verification step is crucial in ensuring the accuracy of our work and provides confidence in the results.

The Nature of the Solutions

In our case, the solutions to the quadratic equation 2t^2 + 2t - 3 = 0 are t = (-1 + √7) / 2 and t = (-1 - √7) / 2. These solutions are real and irrational numbers. The fact that we have two distinct real solutions is evident from the positive discriminant (b^2 - 4ac = 28). A positive discriminant indicates two distinct real roots, a zero discriminant indicates one repeated real root, and a negative discriminant indicates two complex roots. Graphically, these solutions represent the x-intercepts (or roots) of the parabola defined by the quadratic equation y = 2t^2 + 2t - 3. The parabola intersects the x-axis at two points, corresponding to the two solutions we found. Understanding the nature of the solutions helps in visualizing the equation and its behavior, and it connects the algebraic solutions to the geometric representation.

Conclusion

In summary, we have successfully solved the quadratic equation 2t^2 + 2t - 3 = 0 using the quadratic formula. We identified the coefficients, applied the formula, simplified the expression, and verified the solutions. The solutions are t = (-1 + √7) / 2 and t = (-1 - √7) / 2, which are real and irrational. This comprehensive guide has provided a detailed, step-by-step approach to solving quadratic equations, emphasizing the importance of understanding the underlying concepts and verifying the solutions. Mastering these techniques will undoubtedly enhance your problem-solving skills in mathematics and related fields. The quadratic formula is a cornerstone in algebra, and its correct application is vital for numerous mathematical and scientific applications.