Hey, good to see you again. The problem asks to solve the quadratic equation $ax^2 + bx + c = 0$ using the quadratic formula, also known as the "almighty formula". The quadratic formula is given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ We need to consider the discriminant, $\Delta = b^2 - 4ac$, to determine the nature of the roots: If $\Delta > 0$, there are two distinct real roots. If $\Delta = 0$, there is one real root (a repeated root). If $\Delta < 0$, there are two complex conjugate roots. We also need to handle the case where $a=0$, as the equation becomes linear or trivial. 1. Pseudocode Program Step 1: Start the program. Step 2: Declare variables a, b, c for coefficients, discriminant, root1, root2, realPart, imaginaryPart. Step 3: Prompt the user to input the values for a, b, and c. Step 4: Read the values for a, b, and c. Step 5: Check if a is zero. If a == 0: If b == 0: If c == 0: Output "Infinite solutions (0 = 0)". Else (c != 0): Output "No solution (e.g., 5 = 0, which is false)". Else (b != 0): Calculate root = -c / b. Output "This is a linear equation. Root = ", root. Else (a != 0, it's a quadratic equation): Calculate discriminant = bb - 4ac. If discriminant > 0: Calculate root1 = (-b + SQRT(discriminant)) / (2a). Calculate root2 = (-b - SQRT(discriminant)) / (2a). Output "Two distinct real roots: ", root1, " and ", root2. Else If discriminant == 0: Calculate root1 = -b / (2a). Output "One real root (repeated): ", root1. Else (discriminant < 0): Calculate realPart = -b / (2a). Calculate imaginaryPart = SQRT(ABS(discriminant)) / (2a). Output "Two complex conjugate roots: ", realPart, " + ", imaginaryPart, "i", " and ", realPart, " - ", imaginaryPart, "i`. Step 6: End the program. 2. C++ Program `cpp #include <iostream> // Required for input/output operations (cin, cout) #include <cmath> // Required for mathematical functions like sqrt() and abs() int main() { // Declare variables to store the coefficients double a, b, c; // Prompt the user to enter the coefficients std::cout << "Enter coefficient a: "; std::cin >> a; std::cout << "Enter coefficient b: "; std::cin >> b; std::cout << "Enter coefficient c: "; std::cin >> c; // Check if 'a' is zero to handle linear or trivial equations if (a == 0) { if (b == 0) { if (c == 0) { // Case: 0x^2 + 0x + 0 = 0 => 0 = 0 (Infinite solutions) std::cout << "Infinite solutions (0 = 0)." << std::endl; } else { // Case: 0x^2 + 0x + c = 0 (where c != 0) => c = 0 (No solution) std::cout << "No solution (e.g., " << c << " = 0, which is false)." << std::endl; } } else { // Case: 0x^2 + bx + c = 0 (where b != 0) => bx + c = 0 (Linear equation) double root = -c / b; std::cout << "This is a linear equation: " << b << "x + " << c << " = 0." << std::endl; std::cout << "Root = " << root << std::endl; } } else { // If 'a' is not zero, it's a quadratic equation. // Calculate the discriminant double discriminant = b b - 4 a * c; // Determine the nature of the roots based on the discriminant if (discriminant > 0) { // Two distinct real roots double root1 = (-b + sqrt(discriminant)) / (2 * a); double root2 = (-b - sqrt(discriminant)) / (2 * a); std::cout << "Roots are real and different." << std::endl; std::cout << "Root 1 = " << root1 << std::endl; std::cout << "Root 2 = " << root2 << std::endl; } else if (discriminant == 0) { // One real root (repeated) double root1 = -b / (2 * a); std::cout << "Roots are real and same." << std::endl; std::cout << "Root 1 = Root 2 = " << root1 << std::endl; } else { // Two complex conjugate roots double realPart = -b / (2 * a); double imaginaryPart = sqrt(std::abs(discriminant)) / (2 * a); std::cout << "Roots are complex and different." << std::endl; std::cout << "Root 1 = " << realPart << " + " << imaginaryPart << "i" << std::endl; std::cout << "Root 2 = " << realPart << " - " << imaginaryPart << "i" << std::endl; } } return 0; // Indicate successful program execution } ` <br/> What's next? 📸