A quadratic equation is a second-degree polynomial of the form ax2+bx+c=0, where a, b, and c are constants, and a≠0. The solutions to this equation, known as the roots, can be found using the quadratic formula:

x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}

Depending on the discriminant

\Delta = b^2 - 4ac

the equation can have:

- Two distinct real roots (Δ>0)
- One real root (Δ=0)
- Two complex roots (Δ<0)

### Understanding the Quadratic Formula

The quadratic formula provides the roots of the quadratic equation. The discriminant (Δ\DeltaΔ) determines the nature of the roots:

- If Δ>0, the equation has two distinct real roots.
- If Δ=0, the equation as one real root (a repeated root).
- If Δ<0 the equation has two complex roots.

## C++ Program to Find the Roots of a Quadratic Equation

```
#include <iostream>
#include <cmath>
using namespace std;
int main() {
double a, b, c;
cout << "Enter coefficients a, b and c: ";
cin >> a >> b >> c;
if (a == 0) {
cout << "Coefficient 'a' cannot be zero in a quadratic equation." << endl;
return 1; // Exit the program
}
double discriminant = b * b - 4 * a * c;
double sqrt_val = sqrt(abs(discriminant));
if (discriminant > 0) {
cout << "The equation has two distinct real roots:" << endl;
cout << "Root 1: " << (-b + sqrt_val) / (2 * a) << endl;
cout << "Root 2: " << (-b - sqrt_val) / (2 * a) << endl;
} else if (discriminant == 0) {
cout << "The equation has one real root:" << endl;
cout << "Root: " << -b / (2 * a) << endl;
} else {
cout << "The equation has two complex roots:" << endl;
cout << "Root 1: " << -b / (2 * a) << " + " << sqrt_val / (2 * a) << "i" << endl;
cout << "Root 2: " << -b / (2 * a) << " - " << sqrt_val / (2 * a) << "i" << endl;
}
return 0;
}
```

### Output

#### Example 1: Two Distinct Real Roots

```
Enter coefficients a, b and c: 1 -3 2
The equation has two distinct real roots:
Root 1: 2
Root 2: 1
```

#### Example 2: One Real Root

```
Enter coefficients a, b and c: 1 -2 1
The equation has one real root:
Root: 1
```

#### Example 3: Two Complex Roots

```
Enter coefficients a, b and c: 1 2 5
The equation has two complex roots:
Root 1: -1 + 2i
Root 2: -1 - 2i
```

**Explanation:**

**Input Handling:**- The program prompts the user to enter the coefficients a, b, and c.
- It checks if aaa is zero, which would make the equation non-quadratic. If aaa is zero, the program exits with an error message.

**Discriminant Calculation:**- The discriminant is calculated as Δ=b2−4ac.
- The square root of the absolute value of the discriminant is computed for further calculations.

**Root Calculation:**- If the discriminant is positive (Δ>0, the equation has two distinct real roots, calculated and displayed using the quadratic formula.
- If the discriminant is zero (Δ=0), the equation has one real root, calculated and displayed.
- If the discriminant is negative (Δ<0), the equation has two complex roots, calculated and displayed in the form of complex numbers.

## Conclusion

Finding the roots of a quadratic equation using C++ involves understanding the quadratic formula and handling different cases based on the discriminant. This program demonstrates how to implement these concepts in a structured and efficient manner. By using this program, you can quickly and accurately determine the nature and values of the roots for any quadratic equation.