BCA PART A 2. Solve Quadratic Equations

A quadratic equation is a second-order polynomial equation in a single variable, usually written in the standard form:

$�{�}^2+��+�=0$

where $�$ represents the variable, and $�$, $�$, and $�$ are constants with $�\mathrm{\ne }0$. The coefficients $�$, $�$, and $�$ determine the shape and position of the parabola formed by the graph of the quadratic equation.

The solutions to the quadratic equation, also known as the roots, can be found using the quadratic formula:

$�=\frac{-�\pm \sqrt{{�}^2-4��}}{2�}$

The term inside the square root, ${�}^2-4��$, is called the discriminant. The nature of the roots is determined by the value of the discriminant:

• If the discriminant is positive (${�}^2-4��>0$), the equation has two distinct real roots.
• If the discriminant is zero (${�}^2-4��=0$), the equation has one real root (a repeated root).
• If the discriminant is negative (${�}^2-4��<0$), the equation has two complex conjugate roots.

Quadratic equations often arise in various mathematical and scientific contexts, and they are used to model a wide range of physical phenomena. The graph of a quadratic equation is a parabola, and understanding its properties is essential in algebra and calculus.

Program : import cmath # Using cmath for handling complex numbers

def solve_quadratic_equation(a, b, c):

# Calculate the discriminant

delta = cmath.sqrt(b**2 - 4*a*c)

# Calculate the two solutions

root1 = (-b + delta) / (2 * a)

root2 = (-b - delta) / (2 * a)

return root1, root2

# Get user input for coefficients

a = float(input("Enter the coefficient a: "))

b = float(input("Enter the coefficient b: "))

c = float(input("Enter the coefficient c: "))

# Solve the quadratic equation

roots = solve_quadratic_equation(a, b, c)

# Display the roots

print("Root 1:", roots[0])

print("Root 2:", roots[1])

1. Importing cmath:

   • The program starts by importing the cmath module, which provides support for complex numbers. This is necessary because the roots of a quadratic equation may be complex if the discriminant (${�}^2-4��$) is negative.

2. solve_quadratic_equation Function:

   • This function takes three coefficients ($�$, $�$, and $�$) as input and returns the roots of the quadratic equation using the quadratic formula.
   • It calculates the discriminant ($\sqrt{{�}^2-4��}$) using cmath.sqrt.
   • It then calculates the two roots using the quadratic formula and returns them.

3. User Input for Coefficients:

   • The program prompts the user to enter the coefficients $�$, $�$, and $�$ using input().
   • The entered values are converted to floating-point numbers using float().

4. Solving the Quadratic Equation:

   • The program calls the solve_quadratic_equation function with the user-entered coefficients.
   • The roots are stored in the variable roots.

5. Displaying the Roots:

   • The program prints the calculated roots using print().

For example, if the user enters coefficients $�=1$, $�=-3$, and $�=2$, the program will output the roots of the corresponding quadratic equation.