Find 2 Numbers by Sum of Squares and Square of Sum Calculator

Welcome to the free online calculator to find two numbers by given sum of squares and the square of the sum. For example, the sum of the squares of two numbers is 225, and the square of their sum is 441. What are the two numbers?

Let the two numbers be x and y.

According to the question, we can get two equations

x² + y² = 225………(1)

(x + y)² = 441………(2)

Simplify the second equation.

(x + y)² = 441

x² + 2xy + y² = 441………(3)

Subtract the first equation from the third equation

(x² + 2xy + y²) – (x² + y²) = 441 – 225

x² + 2xy + y² – x² – y² = 216

2xy = 216

xy = 108………(4)

Based on Equation 1 and Equation 4 the problem becomes that the product of two numbers is 108 and the sum of their squares is 225. What are the two numbers? How to solve it? There are many ways. For the specific process, please refer to the calculator finds two numbers by product and squares.

Here, the factor pair method is used to solve it.

Since the product of two numbers is 108, these two numbers must be factors of 108, a pair of factors.

Therefore, we can calculate the pair factors of 108 first. Then, substitute them into Equation 1 or 2 to verify.

With the find all factors calculator, we can easily get the pair factors of 108 are

(1, 108), (2, 54), (3, 36), (4, 27), (6, 18) and (9, 12)

Substitute them into Equation 2 respectively.

(1 + 108)² = 11881

(2 + 54)² = 3136

(3 + 36)² = 1521

(4 + 27)² = 961

(6 + 18)² = 576

(9 + 12)² = 441

So, the two numbers are 9 and 12.

Let’s verify.

9² + 12² = 81 + 144 = 225

(9 + 12)² = 441

Correct.

In short, there are many ways to solve the problem, the key is how to choose. If you use our free calculator to find two numbers by the sum of squares and the square of the sum, you can do more with less.

Next, we will introduce how to use the calculator.

Using the calculator is simple, follow these 3 steps:

Next, let’s look at some examples.