# C++ Program - Find Roots of a Quadratic Equation

A standard form of a quadratic quation is:

$ax^2 + bx + c = 0$

Where:

$a, b$ and $c$ are real numbers and $a \ne 0$.

Roots of the equation are:

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

### For Example:

The roots of equation $x^2 + 5x + 4 = 0$ is

$\frac{-5 \pm \sqrt{5^2 - 4\times1\times4}}{2\times1}$ = $\frac{-5 \pm \sqrt{9}}{2}$ = $\frac{-5 \pm 3}{2}$ = $-4, -1$

The roots of the equation will be imaginari if $D = b^2 - 4ac \lt 0$. For example - the roots of equation $x^2 + 4x + 5 = 0$ will be

$\frac{-4 \pm \sqrt{4^2 - 4\times1\times5}}{2\times1}$ = $\frac{-4 \pm \sqrt{-4}}{2}$ = $\frac{-4 \pm 2i}{2}$ = $-2 \pm i$

### Example: Calculate roots of a Quadratic equation

In the below example, a function called roots is created which takes $a$, $b$ and $c$ as arguemts to calculate the roots of the equation $ax^2 + bx + c = 0$.

#include <iostream>
#include <cmath>
using namespace std;

static void roots(double, double, double);

static void roots(double a, double b, double c) {
double D = b*b - 4*a*c;
if (D >= 0){
double x1 = (-b + sqrt(D))/(2*a);
double x2 = (-b - sqrt(D))/(2*a);
cout<<"Roots of the equation ax^2 + bx + c = 0 are :\n";
cout<<x1<<" and "<<x2<<"\n";
cout<<"Where a, b, c are "<<a<<", "<<b<<" and "<<c<<" respectively.\n";
} else {
double x1 = -b/(2*a);
double x2 = sqrt(-D)/(2*a);
cout<<"Roots of the equation ax^2 + bx + c = 0 are imaginary.\n";
cout<<"Real part of root: "<<x1<<"\n";
cout<<"Imaginary part of root: "<<x2<<"\n";
cout<<"Where a, b, c are "<<a<<", "<<b<<" and "<<c<<" respectively.\n";
}
}

int main() {
roots(1,5,4);
cout<<"\n";
roots(1,4,5);
return 0;
}


Output

Roots of the equation ax^2 + bx + c = 0 are :
-1 and -4
Where a, b, c are 1 , 5 and 4 respectively.

Roots of the equation ax^2 + bx + c = 0 are imaginary.
Real part of root: -2
Imaginary part of root: -1
Where a, b, c are 1 , 4 and 5 respectively.