Students are developing a program to find the roots of a quadratic equation given the coefficients of the variables

Answer 1

To find the roots of a quadratic equation given the coefficients, you can use the quadratic formula. A quadratic equation in standard form is expressed as:

\[ ax^2 + bx + c = 0 \]

where:

• \( a \) is the coefficient of \( x^2 \),
• \( b \) is the coefficient of \( x \),
• \( c \) is the constant term.

The roots of the equation can be found using the quadratic formula:

\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]

Step-by-step Implementation

1. Calculate the Discriminant:

   • The discriminant \( D \) is given by \( D = b^2 - 4ac \).
   • If \( D > 0 \), there are two distinct real roots.
   • If \( D = 0 \), there is exactly one real root (a repeated root).
   • If \( D < 0 \), there are two complex roots.

2. Calculate the Roots: Using the quadratic formula, calculate the roots based on the discriminant.

Example Code

Here is a sample Python program that implements the above logic:

# Function to find the roots of a quadratic equation
def find_quadratic_roots(a, b, c):
    # Step 1: Calculate the discriminant
    D = (b ** 2) - (4 * a * c)
    
    # Step 2: Determine the number of roots based on the discriminant
    if D > 0:
        root1 = (-b + D**0.5) / (2 * a)
        root2 = (-b - D**0.5) / (2 * a)
        return (root1, root2)
    elif D == 0:
        root = -b / (2 * a)
        return (root,)
    else:
        realPart = -b / (2 * a)
        imaginaryPart = (-D**0.5) / (2 * a)
        root1 = complex(realPart, imaginaryPart)
        root2 = complex(realPart, -imaginaryPart)
        return (root1, root2)

# Input coefficients a, b, c
a = float(input("Enter coefficient a: "))
b = float(input("Enter coefficient b: "))
c = float(input("Enter coefficient c: "))

# Get the roots
roots = find_quadratic_roots(a, b, c)

# Output the roots
if len(roots) == 2:
    print(f"The roots are: {roots[0]} and {roots[1]}")
elif len(roots) == 1:
    print(f"The root is: {roots[0]}")
else:
    print(f"The roots are: {roots[0]} and {roots[1]}")

Instructions

1. Execute this code in a Python environment.
2. Input the coefficients \( a \), \( b \), and \( c \) when prompted.
3. The program will print the calculated roots based on the input coefficients.

This implementation handles all cases: two distinct real roots, one repeated root, and two complex roots. Make sure to input valid numeric values for the coefficients.