# 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 â€“ (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 roots void findEquation(int A, int B, int C) {     // Print quadratic equation     cout << "(" << C << ")"          << "x^2 +(" << B << ")x + ("          << A << ") = 0"; }   // Driver Code int 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 approach class GFG{    // Function to find the quadratic // equation having reciprocal roots static void findEquation(int A, int B, int C) {           // Print quadratic equation     System.out.print("(" + C + ")" +                  "x^2 +(" + B + ")x + (" +                            A + ") = 0"); }   // Driver Code public 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 roots def findEquation(A, B, C):           # Print quadratic equation     print("(" + str(C)  + ")" +      "x^2 +(" + str(B) + ")x + (" +                 str(A) + ") = 0")   # Driver Code if __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 approach using System; using System.Collections.Generic;   class GFG{    // Function to find the quadratic // equation having reciprocal roots static void findEquation(int A, int B, int C) {     // Print quadratic equation     Console.Write("(" + C + ")" +               "x^2 +(" + B + ")x + (" +                         A + ") = 0"); }   // Driver Code public 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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