Find if two given Quadratic equations have common roots or not

Given values a1, b1 and c1 of first Quadratic equations $a_{1}x^{2} + b_{1}x + c_{1} = 0$ and values a2, b2 and c2 of second Quadratic equations $a_{2}x^{2} + b_{2}x + c_{2} = 0$ , the task is to find whether both quadratic equations have common roots or not.
Examples:

Input: a1 = 1, b1 = -5, c1 = 6, a2 = 2, b2 = -10, c2 = 12
Output: Yes
Explanation:
Roots of both quadratic equations are (2, 3)
Input: a1 = 1, b1 = -5, c1 = 6, a2 = 1, b2 = -9, c2 = 20
Output: No
Explanation:
Roots of first quadratic equations are (2, 3), and Roots of second quadratic equations are (4, 5)
Therefore, both quadratic equations have different roots.

Approach:
Let the two quadratic equations are $a_{1}x^{2} + b_{1}x + c_{1} = 0$ and $a_{2}x^{2} + b_{2}x + c_{2} = 0$

Let us assume the given condition to be true, i.e. both the equations have common roots, say $\alpha$ and $\beta$
As we know that
$\text{Sum of roots = } -\frac{b}{a}$ and $\text{Product of roots = } \frac{c}{a}$
where a, b, c represents the quadratic equation $ax^{2} + bx + c = 0$
Therefore,
For 1st quadratic equation:
$\text{Sum of roots = } \alpha + \beta = -\frac{b_{1}}{a_{1}}$
$\text{Product of roots = } \alpha \beta = \frac{c_{1}}{a_{1}}$
Similarly, for 2nd quadratic equation:
$\text{Sum of roots = } \alpha + \beta = -\frac{b_{2}}{a_{2}}$
$\text{Product of roots = } \alpha \beta = \frac{c_{2}}{a_{2}}$
Now since both the roots are common,
Therefore, from above equations
$\alpha + \beta = -\frac{b_{1}}{a_{1}} = -\frac{b_{2}}{a_{2}}$
$\Rightarrow \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}$
Also,
$\alpha \beta = \frac{c_{1}}{a_{1}} = \frac{c_{2}}{a_{2}}$
$\Rightarrow \frac{c_{1}}{c_{2}} = \frac{a_{1}}{a_{2}}$
Combining the above equations:

which is the required condition for both roots to be common of the two quadratic equations.

Programs:

C++

// C++ Program to Find if two given
// Quadratic equations have
// common roots or not
  
#include <iostream>
using namespace std;
  
// function to check if 2 quadratic
// equations have common roots or not.
bool checkSolution(float a1, float b1,
                   float c1, float a2,
                   float b2, float c2)
{
    return (a1 / a2) == (b1 / b2)
           && (b1 / b2) == (c1 / c2);
}
  
// Driver code
int main()
{
    float a1 = 1, b1 = -5, c1 = 6;
    float a2 = 2, b2 = -10, c2 = 12;
    if (checkSolution(a1, b1, c1, a2, b2, c2))
        cout << "Yes";
    else
        cout << "No";
    return 0;
}

Java

// Java Program to Find if two given
// quadratic equations have common 
// roots or not
class GFG {
      
// Function to check if 2 quadratic
// equations have common roots or not.
static boolean checkSolution(float a1, float b1,
                             float c1, float a2,
                             float b2, float c2)
{
    return ((a1 / a2) == (b1 / b2) && 
            (b1 / b2) == (c1 / c2));
}
      
// Driver code
public static void main (String[] args) 
{
    float a1 = 1, b1 = -5, c1 = 6;
    float a2 = 2, b2 = -10, c2 = 12;
          
    if (checkSolution(a1, b1, c1, a2, b2, c2))
        System.out.println("Yes");
    else
        System.out.println("No");
}
}
  
// This code is contributed by AnkitRai01

Python3

# Python3 program to find if two given 
# quadratic equations have common  
# roots or not 
  
# Function to check if 2 quadratic 
# equations have common roots or not. 
def checkSolution(a1, b1, c1, a2, b2, c2): 
      
    return ((a1 / a2) == (b1 / b2) and 
            (b1 / b2) == (c1 / c2))
  
# Driver code 
a1, b1, c1 = 1, -5, 6
a2, b2, c2 = 2, -10, 12
  
if (checkSolution(a1, b1, c1, a2, b2, c2)): 
    print("Yes") 
else:
    print("No") 
  
# This code is contributed by divyamohan123

C#

// C# Program to Find if two given
// quadratic equations have common 
// roots or not
using System;
class GFG{
      
// Function to check if 2 quadratic
// equations have common roots or not.
static bool checkSolution(float a1, float b1,
                          float c1, float a2,
                          float b2, float c2)
{
    return ((a1 / a2) == (b1 / b2) && 
            (b1 / b2) == (c1 / c2));
}
      
// Driver code
public static void Main (string[] args) 
{
    float a1 = 1, b1 = -5, c1 = 6;
    float a2 = 2, b2 = -10, c2 = 12;
          
    if (checkSolution(a1, b1, c1, a2, b2, c2))
        Console.WriteLine("Yes");
    else
        Console.WriteLine("No");
}
}
  
// This code is contributed by AnkitRai01

Javascript

<script>
  
// Javascript Program to Find if two given
// Quadratic equations have
// common roots or not
  
// function to check if 2 quadratic
// equations have common roots or not.
function checkSolution(a1, b1, c1, a2, b2, c2)
{
    return (a1 / a2) == (b1 / b2)
           && (b1 / b2) == (c1 / c2);
}
  
// Driver code
a1 = 1, b1 = -5, c1 = 6;
a2 = 2, b2 = -10, c2 = 12;
if (checkSolution(a1, b1, c1, a2, b2, c2))
    document.write("Yes");
else
    document.write("No");
  
</script>