Program to find the Roots of a Quadratic Equation

Given a quadratic equation in the form ax² + bx + c, (Only the values of a, b and c are provided) the task is to find the roots of the equation.

Examples:

Input:  a = 1, b = -2, c = 1
Output:  Roots are real and same 1

Input  :  a = 1, b = 7, c = 12
Output:  Roots are real and different
-3, -4

Input  :  a = 1, b = 1, c = 1
Output :  Roots are complex 
-0.5 + i1.73205, -0.5 – i1.73205  

Roots of Quadratic Equation using Sridharacharya Formula:

The roots could be found using the below formula (It is known as the formula of Sridharacharya)

The values of the roots depends on the term (b² – 4ac) which is known as the discriminant (D). 

If D > 0:
        => This occurs when b² > 4ac.
        => The roots are real and unequal.
        => The roots are {-b + ?(b² – 4ac)}/2a and {-b – ?(b² – 4ac)}/2a.

If D = 0:
        => This occurs when b² = 4ac.
        => The roots are real and equal.
        => The roots are (-b/2a).

If D < 0:
        => This occurs when b² < 4ac.
        => The roots are imaginary and unequal.
        => The discriminant can be written as (-1 * -D).
        => As D is negative, -D will be positive.
        => The roots are {-b ± ?(-1*-D)} / 2a = {-b ± i?(-D)} / 2a = {-b ± i?-(b² – 4ac)}/2a where i = ?-1.

Use the following pseudo algorithm to find the roots of the 

Pseudo algorithm:

Start
Read the values of a, b, c
Compute d = b² – 4ac
If d > 0
    calculate root1 = {-b + ?(b² – 4ac)}/2a
    calculate root2 = {-b – ?(b² – 4ac)}/2a
else If d = 0
    calculate root1 = root2 = (-b/2a)
else
    calculate root1 = {-b + i?-(b² – 4ac)}/2a
    calculate root2 = {-b + i?-(b² – 4ac)}/2a
print root1 and root2
End

Below is the implementation of the above formula.

C++

// C++ program to find roots of a quadratic equation
#include <bits/stdc++.h>
using namespace std;

// Prints roots of quadratic equation ax*2 + bx + x
void findRoots(int a, int b, int c)
{
    // If a is 0, then equation is not quadratic, but
    // linear
    if (a == 0) {
        cout << "Invalid";
        return;
    }

    int d = b * b - 4 * a * c;
    double sqrt_val = sqrt(abs(d));

    if (d > 0) {
        cout << "Roots are real and different \n";
        cout << (double)(-b + sqrt_val) / (2 * a) << "\n"
             << (double)(-b - sqrt_val) / (2 * a);
    }
    else if (d == 0) {
        cout << "Roots are real and same \n";
        cout << -(double)b / (2 * a);
    }
    else // d < 0
    {
        cout << "Roots are complex \n";
        cout << -(double)b / (2 * a) << " + i"
             << sqrt_val / (2 * a) << "\n"
             << -(double)b / (2 * a) << " - i"
             << sqrt_val / (2 * a);
    }
}

// Driver code
int main()
{
    int a = 1, b = -7, c = 12;

    // Function call
    findRoots(a, b, c);
    return 0;
}

C

// C program to find roots of a quadratic equation
#include <math.h>
#include <stdio.h>
#include <stdlib.h>

// Prints roots of quadratic equation ax*2 + bx + x
void findRoots(int a, int b, int c)
{
    // If a is 0, then equation is not quadratic, but
    // linear
    if (a == 0) {
        printf("Invalid");
        return;
    }

    int d = b * b - 4 * a * c;
    double sqrt_val = sqrt(abs(d));

    if (d > 0) {
        printf("Roots are real and different \n");
        printf("%f\n%f", (double)(-b + sqrt_val) / (2 * a),
               (double)(-b - sqrt_val) / (2 * a));
    }
    else if (d == 0) {
        printf("Roots are real and same \n");
        printf("%f", -(double)b / (2 * a));
    }
    else // d < 0
    {
        printf("Roots are complex \n");
        printf("%f + i%f\n%f - i%f", -(double)b / (2 * a),
               sqrt_val / (2 * a), -(double)b / (2 * a),
               sqrt_val / (2 * a));
    }
}

// Driver code
int main()
{
    int a = 1, b = -7, c = 12;

    // Function call
    findRoots(a, b, c);
    return 0;
}

Java

// Java program to find roots
// of a quadratic equation

import static java.lang.Math.*;

import java.io.*;
class Quadratic {

    // Prints roots of quadratic
    // equation ax * 2 + bx + x
    static void findRoots(int a, int b, int c)
    {
        // If a is 0, then equation is not
        // quadratic, but linear

        if (a == 0) {
            System.out.println("Invalid");
            return;
        }

        int d = b * b - 4 * a * c;
        double sqrt_val = sqrt(abs(d));

        if (d > 0) {
            System.out.println(
                "Roots are real and different \n");

            System.out.println(
                (double)(-b + sqrt_val) / (2 * a) + "\n"
                + (double)(-b - sqrt_val) / (2 * a));
        }
        else if (d == 0) {
            System.out.println(
                "Roots are real and same \n");

            System.out.println(-(double)b / (2 * a) + "\n"
                               + -(double)b / (2 * a));
        }
        else // d < 0
        {
            System.out.println("Roots are complex \n");

            System.out.println(-(double)b / (2 * a) + " + i"
                               + sqrt_val / (2 * a) + "\n"
                               + -(double)b / (2 * a)
                               + " - i"
                               + sqrt_val / (2 * a));
        }
    }

    // Driver code
    public static void main(String args[])
    {

        int a = 1, b = -7, c = 12;

        // Function call
        findRoots(a, b, c);
    }
}

// This code is contributed by Sumit Kumar.

Python3

# Python program to find roots
# of a quadratic equation
import math

# Prints roots of quadratic equation
# ax*2 + bx + x
def findRoots(a, b, c):

    # If a is 0, then equation is
    # not quadratic, but linear
    if a == 0:
        print("Invalid")
        return -1
    d = b * b - 4 * a * c
    sqrt_val = math.sqrt(abs(d))

    if d > 0:
        print("Roots are real and different ")
        print((-b + sqrt_val)/(2 * a))
        print((-b - sqrt_val)/(2 * a))
    elif d == 0:
        print("Roots are real and same")
        print(-b / (2*a))
    else:  # d<0
        print("Roots are complex")
        print(- b / (2*a), " + i", sqrt_val / (2 * a))
        print(- b / (2*a), " - i", sqrt_val / (2 * a))


# Driver Program
if __name__ == '__main__':
    a = 1
    b = -7
    c = 12

    # Function call
    findRoots(a, b, c)

# This code is contributed by Sharad Bhardwaj.

C#

// C# program to find roots
// of a quadratic equation
using System;

class Quadratic {

    // Prints roots of quadratic
    // equation ax * 2 + bx + x
    void findRoots(int a, int b, int c)
    {

        // If a is 0, then equation is
        // not quadratic, but linear

        if (a == 0) {
            Console.Write("Invalid");
            return;
        }

        int d = b * b - 4 * a * c;
        double sqrt_val = Math.Abs(d);

        if (d > 0) {
            Console.Write(
                "Roots are real and different \n");

            Console.Write(
                (double)(-b + sqrt_val) / (2 * a) + "\n"
                + (double)(-b - sqrt_val) / (2 * a));
        }

        // d < 0
        else {
            Console.Write("Roots are complex \n");

            Console.Write(-(double)b / (2 * a) + " + i"
                          + sqrt_val / (2 * a) + "\n"
                          + -(double)b / (2 * a) + " - i"
                          + sqrt_val / (2 * a));
        }
    }

    // Driver code
    public static void Main()
    {
        Quadratic obj = new Quadratic();
        int a = 1, b = -7, c = 12;

        // Function call
        obj.findRoots(a, b, c);
    }
}

// This code is contributed by nitin mittal.

PHP

<?php
// PHP program to find roots
// of a quadratic equation

// Prints roots of quadratic
// equation ax*2 + bx + x
function findRoots($a, $b, $c)
{
    // If a is 0, then equation is
    // not quadratic, but linear
    if ($a == 0)
    {
        echo "Invalid";
        return;
    }

    $d = $b * $b - 4 * $a * $c;
    $sqrt_val = sqrt(abs($d));

    if ($d > 0)
    {
        echo "Roots are real and ".
                    "different \n";
        echo (-$b + $sqrt_val) / (2 * $a) , "\n",
             (-$b - $sqrt_val) / (2 * $a);
    }
    else if ($d == 0)
    {
        echo "Roots are real and same \n";
        echo -$b / (2 * $a);
    }

    // d < 0
    else
    {
        echo "Roots are complex \n";
        echo -$b / (2 * $a) , " + i" ,
              $sqrt_val / (2 * $a) , "\n" , -$b / (2 * $a),
                             " - i", $sqrt_val / (2 * $a) ;
    }
}

// Driver code
$a = 1; $b = -7 ;$c = 12;

// Function call
findRoots($a, $b, $c);

// This code is contributed
// by nitin mittal.
?>

Javascript

<script>

// JavaScript program to find roots
// of a quadratic equation

    // Prints roots of quadratic
    // equation ax * 2 + bx + x
    function findRoots(a, b, c)
    {
        // If a is 0, then equation is not
        // quadratic, but linear

        if (a == 0) {
            document.write("Invalid");
            return;
        }

        let d = b * b - 4 * a * c;
        let sqrt_val = Math.sqrt(Math.abs(d));

        if (d > 0) {
            document.write(
                "Roots are real and different \n" + "<br/>");

            document.write(
                (-b + sqrt_val) / (2 * a) + "<br/>"
                + (-b - sqrt_val) / (2 * a));
        }
        else if (d == 0) {
            document.write(
                "Roots are real and same \n" + "<br/>");

            document.write(-b / (2 * a) + "<br/>"
                               + -b / (2 * a)) ;
        }
        else // d < 0
        {
            document.write("Roots are complex \n");

            document.write(-b / (2 * a) + " + i"
                               + sqrt_val / (2 * a)  + "<br/>"
                               + -b / (2 * a)
                               + " - i" + sqrt_val) / (2 * a) ;
        }
    }

// Driver Code

        let a = 1, b = -7, c = 12;

        // Function call
        findRoots(a, b, c);

</script>

Output

Roots are real and different 
4.000000
3.000000

Time Complexity: O(log(D)), where D is the discriminant of the given quadratic equation.
Auxiliary Space: O(1)

Self Paced Course

Using Formula in python:

Approach:

• Import the math module for square root and other mathematical operations.
• Declare the coefficients a, b, and c of the quadratic equation.
• Calculate the discriminant discriminant using the formula b^2 – 4ac.
• Check if the discriminant is greater than zero, zero, or less than zero.
• If the discriminant is greater than zero, calculate two real and distinct roots using the formula (-b + sqrt(discriminant)) / (2*a) and (-b – sqrt(discriminant)) / (2*a), and print the roots with the message “Roots are real and distinct”.
• If the discriminant is equal to zero, calculate one real and same root using the formula -b / (2*a), and print the root with the message “Roots are real and same”.
• If the discriminant is less than zero, calculate two complex and different roots using the formula (-b / 2*a) +/- (sqrt(-discriminant) / (2*a)), and print the roots with the message “Roots are complex and different”.

Python3

import math

a = 1
b = -2
c = 1

# Calculate the discriminant
discriminant = b ** 2 - 4 * a * c

if discriminant > 0:
    root1 = (-b + math.sqrt(discriminant)) / (2 * a)
    root2 = (-b - math.sqrt(discriminant)) / (2 * a)
    print("Roots are real and distinct")
    print("Root 1:", root1)
    print("Root 2:", root2)
elif discriminant == 0:
    root = -b / (2 * a)
    print("Roots are real and same")
    print("Root:", root)
else:
    realPart = -b / (2 * a)
    imaginaryPart = math.sqrt(-discriminant) / (2 * a)
    print("Roots are complex and different")
    print("Root 1:", realPart, "+", imaginaryPart, "i")
    print("Root 2:", realPart, "-", imaginaryPart, "i")

Output

Roots are real and same
Root: 1.0

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

Please write comments if you find anything incorrect, or have more information about the topic discussed above.