Round #1072

State the quadratic formula and use it to solve 2x^2 + 5x - 3 = 0. Give the roots as exact rational numbers.

```csharp using System; class Program { static void Main() { // Define the coefficients of the quadratic equation double a = 2; double b = 5; double c = -3; // Calculate the discriminant double discriminant = b * b - 4 * a * c; // Calculate the roots using the quadratic formula double root1 = (-b + Math.Sqrt(discriminant)) / (2 * a); double root2 = (-b - Math.Sqrt(discriminant)) / (2 * a); // Print the roots Console.WriteLine("The roots of the equation are: " + root1 + " and " + root2); } } ```

The quadratic formula is x = (-b ± √(b²-4ac))/(2a), where a, b, and c are coefficients of the quadratic equation ax² + bx + c = 0. To solve the equation 2x^2 + 5x - 3 = 0, we can use the quadratic formula with a = 2, b = 5, and c = -3. ```csharp using System; public class QuadraticSolver { public static void SolveQuadratic(double a, double b, double c) { double discriminant = b * b - 4 * a * c; double root1 = (-b + Math.Sqrt(discriminant)) / (2 * a); double root2 = (-b - Math.Sqrt(discriminant)) / (2 * a); Console.WriteLine($"The roots are {root1} and {root2}"); } public static void Main(string[] args) { SolveQuadratic(2, 5, -3); } } ``` However, since the request asks for exact rational numbers, we need to calculate the roots manually. The discriminant is b²-4ac = 5² - 4*2*(-3) = 25 + 24 = 49. The roots are x = (-b ± √(b²-4ac))/(2a) = (-(5) ± √49)/(2*2) = (-5 ± 7)/4. So, the roots are x = (-5 + 7)/4 = 2/4 = 1/2 and x = (-5 - 7)/4 = -12/4 = -3. Therefore, the roots of the equation 2x^2 + 5x - 3 = 0 are 1/2 and -3.