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Quadratic equation whose roots are K times the roots of given equation
• Last Updated : 05 May, 2021

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

Examples:

Input: A = 1, B = 2, C = 1, K = 2
Output: 1 4 4
Explanation:
The given quadratic equation x2 + 2x + 1 = 0.
Roots of the above equation are -1, -1.
Double of these roots are -2, -2.
Therefore, the quadratic equation with the roots (-2, -2) is x2 + 4x + 4 = 0.

Input: A = 1, B = -7, C = 12, K = 2
Output: 1 -14 48

Approach: The given problem can be solved by using the concept of quadratic roots. Follow the steps below to solve the problem:

• Let the roots of the equation Ax2 + Bx + C = 0 be P and Q respectively.
• Then, the product of the roots of the above equation is given by P * Q = C / A and the sum of the roots of the above equation is given by P + Q = -B / A.
• Therefore, the product of the roots of the required equation is equal to:

(K * P ) * (K * Q) = K2 * P * Q = (K2 * C ) / A

• Similarly, the sum of the roots of the required equation is 2 * K (-B / C).
• Therefore, the required quadratic equation is equal to:

x2 – (Sum of the roots)x + (Product of the roots) = 0

=> Ax2 + (KB)x + (K2)C = 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 whose roots are K times``// the roots of the given equation``void` `findEquation(``int` `A, ``int` `B, ``int` `C,``                  ``int` `K)``{``    ``// Print quadratic equation``    ``cout << A << ``" "` `<< K * B``         ``<< ``" "` `<< K * K * C;``}` `// Driver Code``int` `main()``{``    ``int` `A = 1, B = 2, C = 1, K = 2;` `    ``findEquation(A, B, C, K);` `    ``return` `0;``}`

## Python3

 `# Python3 program for the above approach` `# Function to find the quadratic``# equation whose roots are K times``# the roots of the given equation``def` `findEquation(A, B, C, K):``  ` `    ``# Prquadratic equation``    ``print``(A, K``*``B, K``*``K``*``C)` `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``    ``A, B, C, K ``=` `1``, ``2``, ``1``, ``2` `    ``findEquation(A, B, C, K)` `# This code is contributed by mohit kumar 29.`

## C#

 `// C# program for the above approach``using` `System;` `class` `GFG{` `// Function to find the quadratic``// equation whose roots are K times``// the roots of the given equation``static` `void` `findEquation(``int` `A, ``int` `B,``                         ``int` `C, ``int` `K)``{``    ` `    ``// Print quadratic equation``    ``Console.Write(A + ``" "` `+ K * B +``                      ``" "` `+ K * K * C);``}` `// Driver Code``public` `static` `void` `Main()``{``    ``int` `A = 1, B = 2, C = 1, K = 2;` `    ``findEquation(A, B, C, K);``}``}``    ` `// This code is contributed by ukasp`
Output:
`1 4 4`

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

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