Boundary Value Analysis : Nature of Roots of a Quadratic equation

19 June 2025

1

Consider a problem for the determination of the nature of the roots of a quadratic equation where the inputs are 3 variables (a, b, c) and their values may be from the interval [0, 100]. The output may be one of the following depending on the values of the variables:

Not a quadratic equation,
Real roots,
Imaginary roots,
Equal roots

Our objective is to design the boundary value test cases. Boundary value analysis is a software testing technique in which tests are designed to include representatives of boundary values in a range. A boundary value analysis has a total of 4*n+1 distinct test cases, where n is the number of variables in a problem. Here we have to consider all three variables and design all the distinct possible test cases. We will have a total of 13 test cases as n = 3.

Roots are real if (b² – 4ac) > 0
Roots are imaginary if (b² – 4ac) < 0
Roots are equal if (b² – 4ac) = 0
Equation is not quadratic if a = 0

How do we design the test cases ? For each variable we consider below 5 cases:

a_min = 0
a_min+1 = 1
a_nominal = 50
a_max-1 = 99
a_max = 100

When we are considering these 5 cases for a variable, rest of the variables have the nominal values, like in the above case where the value of ‘a’ is varying from 0 to 100, the value of ‘b’ and ‘c’ will be taken as the nominal or average value. Similarly, when the values of variable ‘b’ are changing from 0 to 100, the values of ‘a’ and ‘c’ will be nominal or average i.e 50. The possible test cases for the nature of roots of a Quadratic Equation in a Boundary Value Analysis can be: Below is the program that verifies the test cases considered in the table shown above. The program takes user-defined inputs so that you can check for any of the test cases mentioned above.

C++

// C++ program to check the nature of the roots
 
#include <bits/stdc++.h>
using namespace std;
 
// BVA for nature of roots of a quadratic equation
void nature_of_roots(int a, int b, int c)
{
 
    // If a = 0, D/2a will yield exception
    // Hence it is not a valid Quadratic Equation
    if (a == 0) {
        cout << "Not a Quadratic Equation"
             << endl;
        return;
    }
 
    int D = b * b - 4 * a * c;
 
    // If D > 0, it will be Real Roots
    if (D > 0) {
        cout << "Real Roots" << endl;
    }
 
    // If D == 0, it will be Equal Roots
    else if (D == 0) {
        cout << "Equal Roots" << endl;
    }
 
    // If D < 0, it will be Imaginary Roots
    else {
        cout << "Imaginary Roots" << endl;
    }
}
 
// Function to check for all testcases
void checkForAllTestCase()
{
 
    cout << "Testcase"
         << "\ta\tb\tc\tActual Output"
         << endl;
    cout << endl;
    int a, b, c;
    int testcase = 1;
    while (testcase <= 13) {
        if (testcase == 1) {
            a = 0;
            b = 50;
            c = 50;
        }
        else if (testcase == 2) {
            a = 1;
            b = 50;
            c = 50;
        }
        else if (testcase == 3) {
            a = 50;
            b = 50;
            c = 50;
        }
        else if (testcase == 4) {
            a = 99;
            b = 50;
            c = 50;
        }
        else if (testcase == 5) {
            a = 100;
            b = 50;
            c = 50;
        }
        else if (testcase == 6) {
            a = 50;
            b = 0;
            c = 50;
        }
        else if (testcase == 7) {
            a = 50;
            b = 1;
            c = 50;
        }
        else if (testcase == 8) {
            a = 50;
            b = 99;
            c = 50;
        }
        else if (testcase == 9) {
            a = 50;
            b = 100;
            c = 50;
        }
        else if (testcase == 10) {
            a = 50;
            b = 50;
            c = 0;
        }
        else if (testcase == 11) {
            a = 50;
            b = 50;
            c = 1;
        }
        else if (testcase == 12) {
            a = 50;
            b = 50;
            c = 99;
        }
        else if (testcase == 13) {
            a = 50;
            b = 50;
            c = 100;
        }
        cout << "\t" << testcase << "\t"
             << a << "\t" << b << "\t"
             << c << "\t";
        nature_of_roots(a, b, c);
        cout << endl;
        testcase++;
    }
}
 
// Driver Code
int main()
{
    checkForAllTestCase();
    return 0;
}

Java

// Java program to check the nature of the roots
import java.util.*;
 
class GFG
{
 
// BVA for nature of roots of a quadratic equation
static void nature_of_roots(int a, int b, int c)
{
 
    // If a = 0, D/2a will yield exception
    // Hence it is not a valid Quadratic Equation
    if (a == 0)
    {
        System.out.print("Not a Quadratic Equation"
            +"\n");
        return;
    }
 
    int D = b * b - 4 * a * c;
 
    // If D > 0, it will be Real Roots
    if (D > 0) {
        System.out.print("Real Roots" +"\n");
    }
 
    // If D == 0, it will be Equal Roots
    else if (D == 0) {
        System.out.print("Equal Roots" +"\n");
    }
 
    // If D < 0, it will be Imaginary Roots
    else {
        System.out.print("Imaginary Roots" +"\n");
    }
}
 
// Function to check for all testcases
static void checkForAllTestCase()
{
 
    System.out.print("Testcase"
        + "\ta\tb\tc\tActual Output"
        +"\n");
    System.out.println();
    int a, b, c;
    a = b = c = 0;
    int testcase = 1;
    while (testcase <= 13) {
        if (testcase == 1) {
            a = 0;
            b = 50;
            c = 50;
        }
        else if (testcase == 2) {
            a = 1;
            b = 50;
            c = 50;
        }
        else if (testcase == 3) {
            a = 50;
            b = 50;
            c = 50;
        }
        else if (testcase == 4) {
            a = 99;
            b = 50;
            c = 50;
        }
        else if (testcase == 5) {
            a = 100;
            b = 50;
            c = 50;
        }
        else if (testcase == 6) {
            a = 50;
            b = 0;
            c = 50;
        }
        else if (testcase == 7) {
            a = 50;
            b = 1;
            c = 50;
        }
        else if (testcase == 8) {
            a = 50;
            b = 99;
            c = 50;
        }
        else if (testcase == 9) {
            a = 50;
            b = 100;
            c = 50;
        }
        else if (testcase == 10) {
            a = 50;
            b = 50;
            c = 0;
        }
        else if (testcase == 11) {
            a = 50;
            b = 50;
            c = 1;
        }
        else if (testcase == 12) {
            a = 50;
            b = 50;
            c = 99;
        }
        else if (testcase == 13) {
            a = 50;
            b = 50;
            c = 100;
        }
        System.out.print("\t" + testcase+ "\t"
            + a+ "\t" + b+ "\t"
            + c+ "\t");
        nature_of_roots(a, b, c);
        System.out.println();
        testcase++;
    }
}
 
// Driver Code
public static void main(String[] args)
{
    checkForAllTestCase();
}
}
 
// This code is contributed by 29AjayKumar

Python3

# Python3 program to check the nature of the roots
 
# BVA for nature of roots of a quadratic equation
def nature_of_roots(a, b, c):
 
    # If a = 0, D/2a will yield exception
    # Hence it is not a valid Quadratic Equation
    if (a == 0):
        print("Not a Quadratic Equation");
        return;
     
    D = b * b - 4 * a * c;
 
    # If D > 0, it will be Real Roots
    if (D > 0):
        print("Real Roots");
     
    # If D == 0, it will be Equal Roots
    elif(D == 0):
        print("Equal Roots");
     
    # If D < 0, it will be Imaginary Roots
    else:
        print("Imaginary Roots");
     
# Function to check for all testcases
def checkForAllTestCase():
 
    print("Testcase\ta\tb\tc\tActual Output");
    print();
    a = b = c = 0;
    testcase = 1;
    while (testcase <= 13):
        if (testcase == 1):
            a = 0;
            b = 50;
            c = 50;
        elif(testcase == 2):
            a = 1;
            b = 50;
            c = 50;
        elif(testcase == 3):
            a = 50;
            b = 50;
            c = 50;
        elif(testcase == 4):
            a = 99;
            b = 50;
            c = 50;
        elif(testcase == 5):
            a = 100;
            b = 50;
            c = 50;
        elif(testcase == 6):
            a = 50;
            b = 0;
            c = 50;
        elif(testcase == 7):
            a = 50;
            b = 1;
            c = 50;
        elif(testcase == 8):
            a = 50;
            b = 99;
            c = 50;
        elif(testcase == 9):
            a = 50;
            b = 100;
            c = 50;
        elif(testcase == 10):
            a = 50;
            b = 50;
            c = 0;
        elif(testcase == 11):
            a = 50;
            b = 50;
            c = 1;
        elif(testcase == 12):
            a = 50;
            b = 50;
            c = 99;
        elif(testcase == 13):
            a = 50;
            b = 50;
            c = 100;
         
        print("\t" , testcase , "\t" , a , "\t" , b , "\t" , c , "\t", end="");
        nature_of_roots(a, b, c);
        print();
        testcase += 1;
     
# Driver Code
if __name__ == '__main__':
    checkForAllTestCase();
 
# This code is contributed by 29AjayKumar

C#

// C# program to check the nature of the roots
using System;
 
class GFG
{
 
// BVA for nature of roots of a quadratic equation
static void nature_of_roots(int a, int b, int c)
{
 
    // If a = 0, D/2a will yield exception
    // Hence it is not a valid Quadratic Equation
    if (a == 0)
    {
        Console.Write("Not a Quadratic Equation"
                       +"\n");
        return;
    }
 
    int D = b * b - 4 * a * c;
 
    // If D > 0, it will be Real Roots
    if (D > 0) {
        Console.Write("Real Roots" +"\n");
    }
 
    // If D == 0, it will be Equal Roots
    else if (D == 0) {
        Console.Write("Equal Roots" +"\n");
    }
 
    // If D < 0, it will be Imaginary Roots
    else {
        Console.Write("Imaginary Roots" +"\n");
    }
}
 
// Function to check for all testcases
static void checkForAllTestCase()
{
 
    Console.Write("Testcase"
        + "\ta\tb\tc\tActual Output"
        +"\n");
    Console.WriteLine();
    int a, b, c;
    a = b = c = 0;
    int testcase = 1;
    while (testcase <= 13) {
        if (testcase == 1) {
            a = 0;
            b = 50;
            c = 50;
        }
        else if (testcase == 2) {
            a = 1;
            b = 50;
            c = 50;
        }
        else if (testcase == 3) {
            a = 50;
            b = 50;
            c = 50;
        }
        else if (testcase == 4) {
            a = 99;
            b = 50;
            c = 50;
        }
        else if (testcase == 5) {
            a = 100;
            b = 50;
            c = 50;
        }
        else if (testcase == 6) {
            a = 50;
            b = 0;
            c = 50;
        }
        else if (testcase == 7) {
            a = 50;
            b = 1;
            c = 50;
        }
        else if (testcase == 8) {
            a = 50;
            b = 99;
            c = 50;
        }
        else if (testcase == 9) {
            a = 50;
            b = 100;
            c = 50;
        }
        else if (testcase == 10) {
            a = 50;
            b = 50;
            c = 0;
        }
        else if (testcase == 11) {
            a = 50;
            b = 50;
            c = 1;
        }
        else if (testcase == 12) {
            a = 50;
            b = 50;
            c = 99;
        }
        else if (testcase == 13) {
            a = 50;
            b = 50;
            c = 100;
        }
        Console.Write("\t" + testcase+ "\t"
                        + a+ "\t" + b+ "\t"
                        + c+ "\t");
        nature_of_roots(a, b, c);
        Console.WriteLine();
        testcase++;
    }
}
 
// Driver Code
public static void Main(String[] args)
{
    checkForAllTestCase();
}
}
 
// This code is contributed by 29AjayKumar

Javascript

// JavaScript program to check the nature of the roots
 
 
// BVA for nature of roots of a quadratic equation
function nature_of_roots(a, b, c)
{
 
    // If a = 0, D/2a will yield exception
    // Hence it is not a valid Quadratic Equation
    if (a == 0) {
        console.log("Not a Quadratic Equation")
        return;
    }
 
    let D = b * b - 4 * a * c;
 
    // If D > 0, it will be Real Roots
    if (D > 0) {
        console.log("Real Roots");
    }
 
    // If D == 0, it will be Equal Roots
    else if (D == 0) {
        console.log("Equal Roots");
    }
 
    // If D < 0, it will be Imaginary Roots
    else {
        console.log("Imaginary Roots");
    }
}
 
// Function to check for all testcases
function checkForAllTestCase()
{
 
    console.log("Testcase\ta\tb\tc\tActual Output\n");
    let a, b, c;
    let testcase = 1;
    while (testcase <= 13) {
        if (testcase == 1) {
            a = 0;
            b = 50;
            c = 50;
        }
        else if (testcase == 2) {
            a = 1;
            b = 50;
            c = 50;
        }
        else if (testcase == 3) {
            a = 50;
            b = 50;
            c = 50;
        }
        else if (testcase == 4) {
            a = 99;
            b = 50;
            c = 50;
        }
        else if (testcase == 5) {
            a = 100;
            b = 50;
            c = 50;
        }
        else if (testcase == 6) {
            a = 50;
            b = 0;
            c = 50;
        }
        else if (testcase == 7) {
            a = 50;
            b = 1;
            c = 50;
        }
        else if (testcase == 8) {
            a = 50;
            b = 99;
            c = 50;
        }
        else if (testcase == 9) {
            a = 50;
            b = 100;
            c = 50;
        }
        else if (testcase == 10) {
            a = 50;
            b = 50;
            c = 0;
        }
        else if (testcase == 11) {
            a = 50;
            b = 50;
            c = 1;
        }
        else if (testcase == 12) {
            a = 50;
            b = 50;
            c = 99;
        }
        else if (testcase == 13) {
            a = 50;
            b = 50;
            c = 100;
        }
        process.stdout.write("\ttestcase\t" + a + "\t" + b + "\t" + c + "\t");
        nature_of_roots(a, b, c);
        testcase++;
    }
}
 
// Driver Code
checkForAllTestCase();
 
 
// This code is contributed bt phasing17

Output:

Testcase    a    b    c    Actual Output

    1    0    50    50    Not a Quadratic Equation

    2    1    50    50    Real Roots

    3    50    50    50    Imaginary Roots

    4    99    50    50    Imaginary Roots

    5    100    50    50    Imaginary Roots

    6    50    0    50    Imaginary Roots

    7    50    1    50    Imaginary Roots

    8    50    99    50    Imaginary Roots

    9    50    100    50    Equal Roots

    10    50    50    0    Real Roots

    11    50    50    1    Real Roots

    12    50    50    99    Imaginary Roots

    13    50    50    100    Imaginary Roots

Time complexity: O(1)
Auxiliary space: O(1)

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Boundary Value Analysis : Nature of Roots of a Quadratic equation

C++

Java

Python3

C#

Javascript

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