Quadratic equation whose roots are reciprocal to the roots of given equation

• Difficulty Level : Basic
• Last Updated : 22 Apr, 2021

Given three integers A, B, and C representing the coefficients of a quadratic equation Ax2 + Bx + C = 0, the task is to find the quadratic equation whose roots are reciprocal to the roots of the given equation.

Examples:

Input: A = 1, B = -5, C = 6
Output: (6)x^2 +(-5)x + (1) = 0
Explanation:
The given quadratic equation x2 – 5x + 6 = 0.
Roots of the above equation are 2, 3.
Reciprocal of these roots are 1/2, 1/3.
Therefore, the quadratic equation with these reciprocal roots is 6x2 – 5x + 1 = 0.

Input: A = 1, B = -7, C = 12
Output: (12)x^2 +(-7)x + (1) = 0

Approach: The idea is to use the concept of quadratic roots to solve the problem. Follow the steps below to solve the problem:

• Consider the roots of the equation Ax2 + Bx + C = 0 to be p, q.
• The product of the roots of the above equation is given by p * q = C / A.
• The sum of the roots of the above equation is given by p + q = -B / A.
• Therefore, the reciprocals of the roots are 1/p, 1/q.
• The product of these reciprocal roots is 1/p * 1/q = A / C.
• The sum of these reciprocal roots is 1/p + 1/q = -B / C.
• If the sum and product of roots is known, the quadratic equation can be x2 – (Sum of the roots)x + (Product of the roots) = 0.
• On solving the above equation, quadratic equation becomes Cx2 + Bx + A = 0.

Below is the implementation of the above approach:

C++

 // C++ program for the above approach #include using namespace std; // Function to find the quadratic// equation having reciprocal rootsvoid findEquation(int A, int B, int C){    // Print quadratic equation    cout << "(" << C << ")"         << "x^2 +(" << B << ")x + ("         << A << ") = 0";} // Driver Codeint main(){    // Given coefficients    int A = 1, B = -5, C = 6;     // Function call to find the quadratic    // equation having reciprocal roots    findEquation(A, B, C);     return 0;}

Java

 // Java program for the above approachclass GFG{  // Function to find the quadratic// equation having reciprocal rootsstatic void findEquation(int A, int B, int C){         // Print quadratic equation    System.out.print("(" + C + ")" +                 "x^2 +(" + B + ")x + (" +                           A + ") = 0");} // Driver Codepublic static void main(String args[]){         // Given coefficients    int A = 1, B = -5, C = 6;     // Function call to find the quadratic    // equation having reciprocal roots    findEquation(A, B, C);}} // This code is contributed by AnkThon

Python3

 # Python3 program for the above approach # Function to find the quadratic# equation having reciprocal rootsdef findEquation(A, B, C):         # Print quadratic equation    print("(" + str(C)  + ")" +     "x^2 +(" + str(B) + ")x + (" +                str(A) + ") = 0") # Driver Codeif __name__ == "__main__":         # Given coefficients    A = 1    B = -5    C = 6     # Function call to find the quadratic    # equation having reciprocal roots    findEquation(A, B, C) # This code is contributed by AnkThon

C#

 // C# program for the above approachusing System;using System.Collections.Generic; class GFG{  // Function to find the quadratic// equation having reciprocal rootsstatic void findEquation(int A, int B, int C){    // Print quadratic equation    Console.Write("(" + C + ")" +              "x^2 +(" + B + ")x + (" +                        A + ") = 0");} // Driver Codepublic static void Main(){         // Given coefficients    int A = 1, B = -5, C = 6;     // Function call to find the quadratic    // equation having reciprocal roots    findEquation(A, B, C);}} // This code is contributed by bgangwar59

Javascript


Output:
(6)x^2 +(-5)x + (1) = 0

Time Complexity: O(1)
Auxiliary Space: O(1)

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