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:
$$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
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
$$\frac{-4 \pm \sqrt{4^2 - 4\times1\times5}}{2\times1}$$ = $$\frac{-4 \pm \sqrt{-4}}{2}$$ = $$\frac{-4 \pm 2i}{2}$$ = $$-2 \pm i$$
In the below example, a method called roots is created which takes $$a$$, $$b$$ and $$c$$ as arguemts to calculate the roots of the equation $$ax^2 + bx + c = 0$$.
using System;
namespace MyApplication {
class MyProgram {
static void roots(double a, double b, double c) {
double D = b*b - 4*a*c;
if (D >= 0){
double x1 = (-b + Math.Sqrt(D))/(2*a);
double x2 = (-b - Math.Sqrt(D))/(2*a);
Console.WriteLine("Roots of the equation ax^2 + bx + c = 0 are :");
Console.WriteLine(x1 + " and " + x2);
Console.WriteLine("Where a, b, c are " + a + "," + b + " and " + c + " respectively.");
} else {
double x1 = -b/(2*a);
double x2 = Math.Sqrt(-D)/(2*a);
Console.WriteLine("Roots of the equation ax^2 + bx + c = 0 are imaginary.");
Console.WriteLine("Real part of root: " + x1);
Console.WriteLine("Imaginary part of root: " + x2);
Console.WriteLine("Where a, b, c are " + a + "," + b + " and " + c + " respectively.");
}
}
static void Main(String[] args) {
roots(1,5,4);
Console.WriteLine();
roots(1,4,5);
}
}
}
Roots of the equation ax^2 + bx + c = 0 are : -1.0 and -4.0 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: -2.0 Imaginary part of root: -1.0 Where a, b, c are 1 , 4 and 5 respectively.
