The standard form of a quadratic equation is:
ax2 + bx + c = 0
Here, a, b, and c are real numbers and a can't be equal to 0.
We can calculate the root of a quadratic by using the formula:
x = (-b ± √(b2-4ac)) / (2a)
The ±
sign indicates that there will be two roots:
root1 = (-b + √(b2-4ac)) / (2a)
root1 = (-b - √(b2-4ac)) / (2a)
The term b2-4ac
is known as the discriminant of a quadratic equation. It specifies the nature of roots. That is,
- if discriminant > 0, roots are real and different
- if discriminant == 0, roots are real and equal
- if discriminant < 0, roots are complex and different
Example: Java Program to Find Roots of a Quadratic Equation
public class Main {
public static void main(String[] args) {
// value a, b, and c
double a = 2.3, b = 4, c = 5.6;
double root1, root2;
// calculate the discriminant (b2 - 4ac)
double discriminant = b * b - 4 * a * c;
// check if discriminant is greater than 0
if (discriminant > 0) {
// two real and distinct roots
root1 = (-b + Math.sqrt(discriminant)) / (2 * a);
root2 = (-b - Math.sqrt(discriminant)) / (2 * a);
System.out.format("root1 = %.2f and root2 = %.2f", root1, root2);
}
// check if discriminant is equal to 0
else if (discriminant == 0) {
// two real and equal roots
// discriminant is equal to 0
// so -b + 0 == -b
root1 = root2 = -b / (2 * a);
System.out.format("root1 = root2 = %.2f;", root1);
}
// if discriminant is less than zero
else {
// roots are complex number and distinct
double real = -b / (2 * a);
double imaginary = Math.sqrt(-discriminant) / (2 * a);
System.out.format("root1 = %.2f+%.2fi", real, imaginary);
System.out.format("\nroot2 = %.2f-%.2fi", real, imaginary);
}
}
}
Output
root1 = -0.87+1.30i and root2 = -0.87-1.30i
In the above program, the coefficients a, b, and c are set to 2.3, 4, and 5.6 respectively. Then, the discriminant
is calculated as b2
- 4ac
.
Based on the value of the discriminant, the roots are calculated as given in the formula above. Notice we've used library function Math.sqrt() to calculate the square root of a number.
We have used the format()
method to print the calculated roots.
The format()
function can also be replaced by printf()
as:
System.out.printf("root1 = root2 = %.2f;", root1);