Number Finder by Sum Square

Input the sum of a number and its square to quickly calculate the number.

Calculate a Number Using the Sum of the Number and Its Square

Result

How to Calculate a Number Using the Sum of the Number and Its Square

Let the number be \( x \), and let the sum of the number and its square be \( S \). The formula begins as: Sum of the number and its square: \( x + x^2 = S \)

Rearrange the formula into a standard quadratic equation: \( x^2 + x - S = 0 \)
Apply the quadratic formula to solve \( x \): \( x = \frac{-1 \pm \sqrt{1 + 4S}}{2} \)

Examples

Example 1: Find the number if the sum of the number and its square is 42.

Solution:

1. Calculate the discriminant:

\(1 + 4S = 1 + 4 \cdot 42 = 169 \)

2. Solve for \( x \):

\(x = \frac{-1 \pm \sqrt{169}}{2} = \frac{-1 \pm 13}{2} \)

\(x = 6 \) or \(x = -7\)

Result: The number is 6 or -7.

Example 2: Find the number if the sum of the number and its square is 110.

Solution:

1. Calculate the discriminant:

\(1 + 4S = 1 + 4 \cdot 110 = 441 \)

2. Solve for \( x \):

\(x = \frac{-1 \pm \sqrt{441}}{2} = \frac{-1 \pm 21}{2} \)

\(x = 10 \) or \(x = -11\)

Result: The number is 10 or -11.