Polynomial Roots

Finding Roots of Quadratics

A quadratic equation $ax^2 + bx + c = 0$ has roots given by the quadratic formula:

x = rac{-b pm sqrt{b^2 - 4ac}}{2a}

The expression under the square root, $Delta = b^2 - 4ac$, is the discriminant:

• $Delta > 0$ → two distinct real roots
• $Delta = 0$ → one repeated real root
• $Delta < 0$ → no real roots (complex)

import math

def quadratic_roots(a, b, c):
    disc = b**2 - 4*a*c
    if disc < 0:
        return None   # complex roots
    x1 = (-b - math.sqrt(disc)) / (2*a)
    x2 = (-b + math.sqrt(disc)) / (2*a)
    return sorted([round(x1, 6), round(x2, 6)])

print(quadratic_roots(1, -3, 2))   # [1.0, 2.0]  (x-1)(x-2)=0
print(quadratic_roots(1, 0, -4))   # [-2.0, 2.0] (x-2)(x+2)=0
print(quadratic_roots(1, 0, 1))    # None (complex roots)

Applications

• Signal processing — filter design uses polynomial roots
• Control systems — stability determined by root locations
• Numerical methods — root-finding is fundamental to scientific computing

Your Task

Implement quadratic_roots(a, b, c) that returns a sorted list of real roots [x1, x2] rounded to 6 decimal places, or None if the roots are complex (discriminant $< 0$).