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

Given three integers **A, B**, and **C** representing the coefficients of a quadratic equation **Ax ^{2} + Bx + C = 0**, the task is to find the quadratic equation whose roots are reciprocal to the roots of the given equation.

**Examples:**

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Input:A = 1, B = -5, C = 6Output:(6)x^2 +(-5)x + (1) = 0Explanation:

The given quadratic equation x^{2}– 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 6x^{2}– 5x + 1 = 0.

Input:A = 1, B = -7, C = 12Output:(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
**Ax**to be^{2}+ Bx + C = 0**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
**x**^{2}â€“ (Sum of the roots)x + (Product of the roots) = 0. - On solving the above equation, quadratic equation becomes
**Cx**^{2}+ Bx + A = 0.

Below is the implementation of the above approach:

## C++

`// C++ program for the above approach` `#include <bits/stdc++.h>` `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

`<script>` ` ` `// Javascript program for the above approach` ` ` `// Function to find the quadratic` ` ` `// equation having reciprocal roots` ` ` `function` `findEquation(A, B, C) ` ` ` `{` ` ` `// Print quadratic equation` ` ` `document.write(` `"("` `+ C + ` `")"` `+` ` ` `"x^2 +("` `+ B +` ` ` `")x + ("` `+ A + ` `") = 0"` `)` ` ` `}` ` ` `// Driver Code` ` ` `// Given coefficients` ` ` `let 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 Hritik` ` ` ` ` `</script>` |

**Output:**

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

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