Solving 3x² - 7x - 5 = 0 Exact And Approximate Solutions

In this article, we delve into the solution of the quadratic equation 3x² - 7x - 5 = 0. Quadratic equations, characterized by their highest power of 2, play a pivotal role in various mathematical and scientific applications. Understanding how to solve them is crucial for anyone venturing into fields like physics, engineering, economics, and computer science. This guide offers a step-by-step approach to finding both exact and approximate solutions, employing the well-known quadratic formula. By the end of this guide, you will not only be able to solve this specific equation but also gain a deeper understanding of quadratic equations in general.

The quadratic equation we aim to solve is 3x² - 7x - 5 = 0. This equation is in the standard form of a quadratic equation, which is ax² + bx + c = 0, where a, b, and c are constants. In our case, a = 3, b = -7, and c = -5. The solutions to this equation, also known as roots or zeros, are the values of x that satisfy the equation. These solutions can be real or complex numbers, and they represent the points where the parabola described by the quadratic equation intersects the x-axis. Finding these solutions is a fundamental problem in algebra, and the quadratic formula provides a powerful method for doing so.

Before diving into the quadratic formula, it’s worth noting that there are other methods for solving quadratic equations, such as factoring and completing the square. However, the quadratic formula is particularly useful because it works for any quadratic equation, regardless of whether it can be easily factored. This makes it a versatile tool in our mathematical arsenal. In the following sections, we will first present the quadratic formula, then apply it to our specific equation, and finally discuss the nature of the solutions we obtain. We will also explore how to express these solutions in both exact and approximate forms, providing a comprehensive understanding of the equation’s roots.

The quadratic formula is a cornerstone of algebra, providing a direct method to find the solutions (roots) of any quadratic equation. A quadratic equation is generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The quadratic formula is derived by completing the square on the general quadratic equation, and it states that the solutions for x are given by:

x = (-b ± √(b² - 4ac)) / (2a)

This formula is essential for solving quadratic equations because it works for all possible values of a, b, and c, whether they are integers, fractions, or even irrational numbers. The ± symbol indicates that there are generally two solutions: one obtained by adding the square root term and the other by subtracting it. These solutions may be real or complex, depending on the value of the discriminant, which is the expression inside the square root (b² - 4ac).

The discriminant plays a crucial role in determining the nature of the solutions. If the discriminant is positive (b² - 4ac > 0), the equation has two distinct real solutions. This means that the parabola described by the quadratic equation intersects the x-axis at two different points. If the discriminant is zero (b² - 4ac = 0), the equation has exactly one real solution, which is a repeated root. In this case, the parabola touches the x-axis at a single point. If the discriminant is negative (b² - 4ac < 0), the equation has two complex solutions, which are complex conjugates of each other. This means that the parabola does not intersect the x-axis at any real points.

The quadratic formula is not only a powerful tool for finding solutions but also a testament to the beauty and elegance of mathematical formulas. Its derivation involves clever algebraic manipulations, and its application is straightforward once the values of a, b, and c are identified. In the next section, we will apply this formula to our specific equation, 3x² - 7x - 5 = 0, to find its exact solutions. Understanding the quadratic formula is a fundamental skill in algebra, and mastering its use will greatly enhance your ability to solve a wide range of mathematical problems.

To solve the quadratic equation 3x² - 7x - 5 = 0, we will meticulously apply the quadratic formula. As established earlier, the quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

In our equation, 3x² - 7x - 5 = 0, we identify the coefficients as follows: a = 3, b = -7, and c = -5. Now, we substitute these values into the quadratic formula to find the solutions for x.

First, let's substitute the values into the formula:

x = (-(-7) ± √((-7)² - 4 * 3 * (-5))) / (2 * 3)

Next, we simplify the expression step by step. The negation of -7 is 7, so we have:

x = (7 ± √(49 - 4 * 3 * (-5))) / (6)

Now, we calculate the value inside the square root. 4 * 3 * (-5) equals -60, so we have:

x = (7 ± √(49 - (-60))) / (6)

Subtracting a negative number is the same as adding its positive counterpart, so we get:

x = (7 ± √(49 + 60)) / (6)

Adding 49 and 60 gives us 109, so the expression becomes:

x = (7 ± √109) / (6)

At this point, we have arrived at the exact solutions for the quadratic equation. The solutions are (7 + √109) / 6 and (7 - √109) / 6. These are the exact solutions because they involve the square root of 109, which is an irrational number. Therefore, these solutions cannot be expressed as simple fractions or terminating decimals. The exact solutions provide a precise representation of the roots of the equation, which is essential in many theoretical and practical applications.

In the next section, we will approximate these solutions to obtain decimal values. This will give us a better sense of the numerical magnitude of the roots and make them easier to use in real-world scenarios where an approximate value is sufficient. Understanding how to both find the exact solutions and approximate them is a valuable skill in mathematics and its applications.

The exact solutions to the quadratic equation 3x² - 7x - 5 = 0, as derived using the quadratic formula, are given by:

x = (7 + √109) / 6

and

x = (7 - √109) / 6

These solutions are considered exact because they represent the roots of the equation in their most precise form, involving the square root of 109 (√109). The square root of 109 is an irrational number, meaning it cannot be expressed as a simple fraction or a terminating decimal. Therefore, the exact solutions preserve the full accuracy of the roots without any rounding or approximation. This is particularly important in mathematical contexts where precision is paramount, such as in theoretical calculations, scientific research, and engineering design.

To further understand the significance of these exact solutions, let's consider the nature of √109. It is a non-repeating, non-terminating decimal, which means its decimal representation goes on infinitely without any repeating pattern. When we include √109 in our solutions, we maintain this infinite precision. If we were to approximate √109 with a decimal value, we would inevitably introduce some degree of error. While this error might be small, it can accumulate in more complex calculations, potentially leading to inaccurate results.

The exact solutions also provide a clear and concise way to represent the roots algebraically. This is particularly useful when dealing with further algebraic manipulations or when comparing solutions to other equations. For example, if we need to perform operations involving these roots, such as adding or multiplying them, using the exact solutions allows us to work with precise values and avoid the complications that can arise from approximations.

In many applications, however, approximate solutions are sufficient and more practical to use. In the next section, we will explore how to approximate these exact solutions to obtain decimal values. This will involve using a calculator to find the decimal representation of √109 and then performing the necessary arithmetic operations. Understanding both the exact and approximate solutions provides a comprehensive understanding of the roots of the quadratic equation and allows us to choose the representation that is most appropriate for the task at hand.

To obtain the approximate solutions for the quadratic equation 3x² - 7x - 5 = 0, we need to convert the exact solutions (7 ± √109) / 6 into decimal representations. This involves using a calculator to find the decimal value of √109 and then performing the necessary arithmetic operations. Approximating the solutions gives us a clearer understanding of their numerical magnitude and makes them easier to use in practical applications where an exact value is not required.

First, we find the approximate value of √109 using a calculator. The square root of 109 is approximately 10.4403. Now, we can substitute this value into our exact solutions:

For the first solution, (7 + √109) / 6, we have:

x ≈ (7 + 10.4403) / 6

Adding 7 and 10.4403 gives us 17.4403, so:

x ≈ 17.4403 / 6

Dividing 17.4403 by 6, we get:

x ≈ 2.9067

So, the first approximate solution is approximately 2.9067.

For the second solution, (7 - √109) / 6, we have:

x ≈ (7 - 10.4403) / 6

Subtracting 10.4403 from 7 gives us -3.4403, so:

x ≈ -3.4403 / 6

Dividing -3.4403 by 6, we get:

x ≈ -0.5734

So, the second approximate solution is approximately -0.5734.

Therefore, the approximate solutions to the quadratic equation 3x² - 7x - 5 = 0 are approximately 2.9067 and -0.5734. These decimal values provide a practical way to visualize the roots of the equation on a number line and use them in calculations where precision to several decimal places is sufficient. In many real-world applications, such as engineering and physics, approximate solutions are commonly used because they provide a balance between accuracy and ease of computation.

It’s important to note that the level of approximation can be adjusted depending on the required precision. For example, we could round the solutions to two decimal places (2.91 and -0.57) or even to the nearest whole number (3 and -1) if less precision is needed. The key is to understand the context in which the solutions will be used and choose the level of approximation that is most appropriate.

In conclusion, we have successfully solved the quadratic equation 3x² - 7x - 5 = 0 using the quadratic formula. We found both the exact solutions, (7 + √109) / 6 and (7 - √109) / 6, and the approximate solutions, which are approximately 2.9067 and -0.5734. This process has highlighted the importance of understanding and applying the quadratic formula, a fundamental tool in algebra.

The exact solutions provide the most precise representation of the roots of the equation, preserving the full accuracy of the irrational number √109. These solutions are essential in theoretical contexts where precision is paramount. The approximate solutions, on the other hand, offer a practical way to understand the numerical magnitude of the roots and use them in real-world applications where a decimal representation is more convenient.

Throughout this guide, we have emphasized the step-by-step application of the quadratic formula, from identifying the coefficients a, b, and c to simplifying the expression and obtaining the solutions. We have also discussed the significance of the discriminant in determining the nature of the solutions and the importance of both exact and approximate representations.

Solving quadratic equations is a crucial skill in mathematics and its applications. The quadratic formula provides a versatile and reliable method for finding the roots of any quadratic equation, regardless of whether it can be easily factored or not. By mastering this technique, you will be well-equipped to tackle a wide range of mathematical problems in various fields.

This comprehensive guide has not only provided the solutions to the specific equation 3x² - 7x - 5 = 0 but also aimed to enhance your understanding of quadratic equations in general. We hope that this knowledge will serve as a solid foundation for your future mathematical endeavors.