Question

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:

```python
# 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.