Are you really going to solve 2024x^2+2023x-1=0 using delta formula?

We all know that we can solve any quadratic equation using quadratic formula which is:

But try to solve this using quadratic formula:

Of course numbers are big and makes calculation very hard but I wanna show you two tricks to solve these equations very fast!

1. if a+b+c=0, one root is 1 and another is c/a

2. if a+c=b, one root -1 and another is -c/a

In this equation we see that a+c is b, 2023+1=2024 so one root is -1 and another is -1/2023, so fast right?

but where does these tricks come from? well we need to know one thing, sum of two roots is -b/a:

As you can see the deltas are canceled and we are left with -b/a

now consider a+b+c=0

add -b to whole equation and devide it by a:

a+c=-b

a/a + c/a = -b/a

1 + c/a = -b/a

intresting, we can see the -b/a which is sum of the roots so one root is 1 and another is c/a

now consider a+c=b

multiply -1 to whole equation and divid it by a:

-a-c=-b

-a/a - c/a = -b/a

-1 - c/a = -b/a

again we see the -b/a, so one root is -1 and another is -c/a

I have made an app in java so you can calculate quadratic equations using it:

import java.util.InputMismatchException;
import java.util.Scanner;

public class Main {
    public static void main(String[] args) {
        var main = new Main();

        final var scanner = new Scanner(System.in);
        while (true) {
            System.out.println("Enter e to exit or c to continue");
            if (scanner.next().equals("e")) {
                scanner.close();
                break;
            }
            try {
                System.out.println("Enter a, b and c of the equation: ");
                final var a = scanner.nextDouble();
                final var b = scanner.nextDouble();
                final var c = scanner.nextDouble();
                main.solveEquation(a, b, c);
            }catch (InputMismatchException e) {
                System.out.println("Input is invalid");
                scanner.next();
            }
        }
    }

    public void solveEquation(double a, double b, double c) {
        final var delta = Math.pow(b, 2) - (4*a*c);

        if (delta < 0) {
            System.out.println("Equation has imaginary results");
            final var x1 = String.format("(%s + %si)/%s", -b, Math.sqrt(Math.abs(delta)), 2 * a);
            final var x2 = String.format("(%s - %si)/%s", -b, Math.sqrt(Math.abs(delta)), 2 * a);
            System.out.printf("x1: %s\nx2: %s\n", x1, x2);
            return;
        }

        final var x1 = (-b + Math.sqrt(delta)) / (2*a);
        final var x2 = (-b - Math.sqrt(delta)) / (2*a);
        System.out.printf("x1: %s\nx2: %s\n", x1, x2);
    }
}