Quadratic equation whose roots are K times the roots of given equation
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.
Input: A = 1, B = 2, C = 1, K = 2
Output: 1 4 4
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:
1 4 4
Time Complexity: O(1)
Auxiliary Space: O(1)