What is the optimal time complexity of solving Finding Roots of a quadratic equation?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for Finding Roots of a quadratic equation?

The space complexity is O(1) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in Finding Roots of a quadratic equation?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for Finding Roots of a quadratic equation?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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Python BasicsEasy

Finding Roots of a quadratic equation

Learn how to solve the 'Finding Roots of a quadratic equation' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Easy

Write a function find_roots(a, b, c) that takes three numbers a, b, and c representing the coefficients of a quadratic equation ax² + bx + c = 0, and returns the roots.

Use the quadratic formula: x = (-b ± √(b²-4ac)) / (2a)

Return:

- A tuple of two floats (root1, root2) rounded to 2 decimal places when b²-4ac >= 0 (root1 <= root2)

- The string 'Complex Roots' when b²-4ac < 0 (discriminant is negative)

Assume a ≠ 0.

Constraints

•-1000 <= a, b, c <= 1000
•a != 0

Examples

Example 1

Input

find_roots(1, -3, 2)

Output

(1.0, 2.0)

Explanation

x² - 3x + 2 = 0. Discriminant = 9 - 8 = 1. Roots: (3±1)/2 = 2 and 1. Sorted: (1.0, 2.0).

Example 2

Input

find_roots(1, -2, 1)

Output

(1.0, 1.0)

Explanation

x² - 2x + 1 = 0. Discriminant = 4 - 4 = 0. Double root: x = 2/2 = 1.0.

Example 3

Input

find_roots(1, 1, 1)

Output

'Complex Roots'

Explanation

x² + x + 1 = 0. Discriminant = 1 - 4 = -3 < 0. No real roots.

Need a Hint?

Use simple arithmetic operators (like modulo `%`, division `//`), conditional checks, or loops to inspect number properties.

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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