Take two numbers m and n & return two numbers whose sum is n and product m in JavaScript

We need to write a JavaScript function that takes two numbers m (product) and n (sum) and returns two numbers whose sum equals n and product equals m. If no such numbers exist, the function should return false.

This is essentially solving a quadratic equation where we need to find two numbers x and y such that x + y = n and x × y = m.

Mathematical Approach

We can solve this using the quadratic formula. If x + y = n and x × y = m, then x and y are roots of the equation t² - nt + m = 0.

Example

const findNumbers = (product, sum) => {
    // Using quadratic formula to find roots of t² - sum*t + product = 0
    const discriminant = sum * sum - 4 * product;
    
    // If discriminant is negative, no real solutions exist
    if (discriminant < 0) {
        return false;
    }
    
    const sqrtDiscriminant = Math.sqrt(discriminant);
    const x = (sum + sqrtDiscriminant) / 2;
    const y = (sum - sqrtDiscriminant) / 2;
    
    // Check if solutions are valid (avoiding floating point errors)
    if (Math.abs(x + y - sum) < 1e-10 && Math.abs(x * y - product) < 1e-10) {
        return [x, y];
    }
    
    return false;
};

console.log(findNumbers(144, 24));  // Product: 144, Sum: 24
console.log(findNumbers(45, 14));   // Product: 45, Sum: 14  
console.log(findNumbers(98, 21));   // Product: 98, Sum: 21
console.log(findNumbers(10, 3));    // Product: 10, Sum: 3 (impossible)

[ 12, 12 ]
[ 5, 9 ]
[ 7, 14 ]
false

Alternative Brute Force Approach

For simpler cases or when dealing with integers, we can use a brute force method:

const findNumbersBruteForce = (product, sum) => {
    for (let i = 0; i <= sum / 2; i++) {
        const j = sum - i;
        if (i * j === product) {
            return [i, j];
        }
    }
    return false;
};

console.log(findNumbersBruteForce(144, 24));
console.log(findNumbersBruteForce(45, 14));
console.log(findNumbersBruteForce(98, 21));
console.log(findNumbersBruteForce(10, 3));

[ 12, 12 ]
[ 5, 9 ]
[ 7, 14 ]
false

Comparison

Key Points

• The quadratic formula approach is more efficient with O(1) time complexity
• Check the discriminant to determine if real solutions exist
• Use floating point comparison for accuracy when dealing with decimal results
• The brute force method works well for integer solutions

Conclusion

Both approaches solve the problem effectively. Use the quadratic formula for mathematical precision and efficiency, or the brute force method for simpler integer-based solutions.