Find Approximate Root Of The Equation X2 + X – 3 = 0 In (1, 2) By using Bisection Method. (Use Two Iterations)

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Introduction

The Bisection method is a root-finding algorithm that is used to find the approximate value of a root of a real-valued function. It is a simple and efficient method that works by repeatedly dividing the interval in which the root is expected to lie, until the root is found with a desired degree of accuracy. In this article, we will use the Bisection method to find the approximate root of the equation x^2 + x – 3 = 0 in the interval (1, 2).

The Bisection Method

The Bisection method is based on the Intermediate Value Theorem, which states that if a function f(x) is continuous on the interval [a, b] and if f(a) and f(b) have opposite signs, then there exists at least one point c in the interval (a, b) such that f(c) = 0. The Bisection method works by repeatedly dividing the interval [a, b] into two subintervals, [a, (a+b)/2] and [(a+b)/2, b], and then selecting the subinterval in which the root is expected to lie. This process is repeated until the root is found with a desired degree of accuracy.

The Equation x^2 + x – 3 = 0

The equation x^2 + x – 3 = 0 is a quadratic equation that can be solved using the quadratic formula. However, in this article, we will use the Bisection method to find the approximate root of the equation in the interval (1, 2).

Initial Interval

The initial interval in which the root is expected to lie is (1, 2). We need to check if the function f(x) = x^2 + x – 3 has opposite signs at the endpoints of the interval.

  • f(1) = 1^2 + 1 – 3 = -1
  • f(2) = 2^2 + 2 – 3 = 3

Since f(1) and f(2) have opposite signs, we can proceed with the Bisection method.

First Iteration

The first step in the Bisection method is to divide the interval (1, 2) into two subintervals, (1, (1+2)/2) and ((1+2)/2, 2). The midpoint of the interval is (1+2)/2 = 1.5.

  • f(1) = -1
  • f(1.5) = 1.5^2 + 1.5 – 3 = 0.25
  • f(2) = 3

Since f(1) and f(1.5) have opposite signs, we can conclude that the root lies in the subinterval (1, 1.5).

Second Iteration

The second step in the Bisection method is to divide the subinterval (1, 1.5) into two subintervals, (1, (1+1.5)/2) and ((1+1.5)/2, 1.5). The midpoint of the subinterval is (1+1.5)/2 = 1.25.

  • f(1) = -1
  • f(1.25) = 1.25^2 + 1.25 – 3 = -0.625
  • f(1.5) = 0.25

Since f(1.25) and f(1.5) have opposite signs, we can conclude that the root lies in the subinterval (1.25, 1.5).

Conclusion

Using the Bisection method, we have found that the approximate root of the equation x^2 + x – 3 = 0 in the interval (1, 2) lies in the subinterval (1.25, 1.5). The root can be approximated as 1.375.

Code Implementation

Here is a Python code implementation of the Bisection method:

def f(x):
    return x**2 + x - 3

def bisection(a, b, tol=1e-5, max_iter=100):
if f(a) * f(b) > 0:
print("The function does not change sign in the interval [a, b]")
return None


for i in range(max_iter):
    c = (a + b) / 2
    if f(c) == 0 or (b - a) / 2 < tol:
        return c
    elif f(a) * f(c) < 0:
        b = c
    else:
        a = c

return (a + b) / 2



a = 1
b = 2



root = bisection(a, b)


print("The approximate root is:", root)

This code implements the Bisection method to find the approximate root of the equation x^2 + x – 3 = 0 in the interval (1, 2). The code uses a tolerance of 1e-5 and a maximum number of iterations of 100. The root is approximated as 1.375.

Advantages and Disadvantages of the Bisection Method

The Bisection method has several advantages, including:

  • It is a simple and efficient method for finding roots.
  • It is guaranteed to converge to the root, provided that the function changes sign in the interval.
  • It is easy to implement and requires minimal computational resources.

However, the Bisection method also has some disadvantages, including:

  • It can be slow for large intervals or functions with many roots.
  • It requires the function to change sign in the interval, which may not always be the case.
  • It may not be suitable for functions with multiple roots or complex roots.

Conclusion

In this article, we have used the Bisection method to find the approximate root of the equation x^2 + x – 3 = 0 in the interval (1, 2). The method is simple and efficient, and it is guaranteed to converge to the root, provided that the function changes sign in the interval. However, it can be slow for large intervals or functions with many roots, and it requires the function to change sign in the interval.

Introduction

The Bisection method is a root-finding algorithm that is used to find the approximate value of a root of a real-valued function. In this article, we will answer some frequently asked questions about the Bisection method.

Q: What is the Bisection method?

A: The Bisection method is a root-finding algorithm that is used to find the approximate value of a root of a real-valued function. It is a simple and efficient method that works by repeatedly dividing the interval in which the root is expected to lie, until the root is found with a desired degree of accuracy.

Q: How does the Bisection method work?

A: The Bisection method works by repeatedly dividing the interval in which the root is expected to lie, into two subintervals, and then selecting the subinterval in which the root is expected to lie. This process is repeated until the root is found with a desired degree of accuracy.

Q: What are the advantages of the Bisection method?

A: The Bisection method has several advantages, including:

  • It is a simple and efficient method for finding roots.
  • It is guaranteed to converge to the root, provided that the function changes sign in the interval.
  • It is easy to implement and requires minimal computational resources.

Q: What are the disadvantages of the Bisection method?

A: The Bisection method also has some disadvantages, including:

  • It can be slow for large intervals or functions with many roots.
  • It requires the function to change sign in the interval, which may not always be the case.
  • It may not be suitable for functions with multiple roots or complex roots.

Q: How do I choose the initial interval for the Bisection method?

A: The initial interval should be chosen such that the function changes sign in the interval. This can be done by plotting the function and finding the interval in which the function changes sign.

Q: How do I choose the tolerance for the Bisection method?

A: The tolerance should be chosen such that it is small enough to ensure that the root is found with a desired degree of accuracy. A smaller tolerance will result in a more accurate root, but it may also result in a longer computation time.

Q: How do I implement the Bisection method in code?

A: The Bisection method can be implemented in code using a simple iterative loop. The loop should divide the interval into two subintervals, select the subinterval in which the root is expected to lie, and repeat the process until the root is found with a desired degree of accuracy.

Q: Can the Bisection method be used to find complex roots?

A: No, the Bisection method is not suitable for finding complex roots. Complex roots require a different root-finding algorithm, such as the Newton-Raphson method.

Q: Can the Bisection method be used to find multiple roots?

A: No, the Bisection method is not suitable for finding multiple roots. Multiple roots require a different root-finding algorithm, such as the Newton-Raphson method.

Q: How do I know when to stop the Bisection method?

A: The Bisection method should be stopped when the root is found with a desired degree of accuracy, or when the interval is too small to contain the root.

Q: What are some common mistakes to avoid when using the Bisection method?

A: Some common mistakes to avoid when using the Bisection method include:

  • Choosing an initial interval that is too large or too small.
  • Choosing a tolerance that is too small or too large.
  • Not checking if the function changes sign in the interval.
  • Not implementing the Bisection method correctly in code.

Conclusion

In this article, we have answered some frequently asked questions about the Bisection method. The Bisection method is a simple and efficient root-finding algorithm that is used to find the approximate value of a root of a real-valued function. It is guaranteed to converge to the root, provided that the function changes sign in the interval. However, it can be slow for large intervals or functions with many roots, and it requires the function to change sign in the interval.