find the roots of a quadratic using the quadratic formula

certainly! finding the roots of a quadratic equation is a common task in algebra. a quadratic equation is generally represented in the form:

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

where:
- \( a \), \( b \), and \( c \) are constants (with \( a \neq 0 \)).
- \( x \) represents the variable.

### quadratic formula

the roots (or solutions) of a quadratic equation can be found using the **quadratic formula**:

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

where:
- the expression \( b^2 - 4ac \) is known as the **discriminant**.
- the discriminant determines the nature of the roots:
- if \( b^2 - 4ac 0 \): two distinct real roots.
- if \( b^2 - 4ac = 0 \): one real root (a repeated root).
- if \( b^2 - 4ac 0 \): no real roots (the roots are complex).

### steps to find the roots

1. identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation.
2. calculate the discriminant \( d = b^2 - 4ac \).
3. use the quadratic formula to find the roots based on the value of the discriminant.

### python code example

here’s a simple python program that implements the quadratic formula:

### explanation of the code

1. **imports**: the `cmath` module is imported for its ability to handle complex numbers, which is useful when the discriminant is negative.

2. **function definition**: `find_roots(a, b, c)` is defined to take the coefficients of the quadratic equation as parameters.

3. **discriminant calculation**: the discriminant \( d \) is calculated as \( b^2 - 4ac \).

5. **example usage**: the program defines values for \( a \), \( b \), and \( c \), calls the `find_roots` function, and prints the results.

### running the code

you can run this code snippet in any python environment. change the values of \( a \), \( b \), and \( ...

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