Find Two Numbers by Product and Squares Calculator

Welcome to the free online calculator to find two numbers by product and sum of squares. For example, the product of two numbers is 108 and the sum of their squares is 225. What are the numbers?

Assuming these two numbers are x and y.

Then, we can get two equations based on the above problem description.

x * y = 108………(1)

x² + y² = 225………(2)

Multiply both sides of the first equation by 2 to get the third equation.

2xy = 216………(3)

Next, combine the second equation with the third equation to get two new equations.

Add two equations

x² + y² + 2xy = 225 + 216

(x + y)² = 441

x + y = 21………(4)

Subtract two equations

x² + y² – 2xy = 225 – 216

(x – y)² = 9

x – y = 3………(5)

According to these few equations, the problem can be transformed into diversity.

For example

Combine the first and the 4th equations

xy=108

x + y =21

The problem becomes: the sum of two numbers is 21 and their product is 108. What are the numbers? It can be solved with find two numbers by sum and product calculator.

Combine the first and the 5th equations

xy=108

x – y =3

The problem becomes: the difference between two numbers is 3 and their product is 108. What are the numbers? It can be solved with find two numbers by difference and product calculator.

Combine the second and the 4th equations

x² + y² = 225

x + y =21

The problem becomes: the sum of two numbers is 21 and the sum of their squares is 225. What are the numbers? It can be solved with find two numbers by sum and squares calculator.

Combine the second and the 5th equations

x² + y² = 225

x – y =3

The problem becomes: the difference between two numbers is 3 and the sum of their squares is 225. What are the numbers? It can be solved with find two numbers by difference and squares calculator.

Of course, the simplest is the combination of the 4th equation and the 5th equation.

x + y = 21

x – y = 3

The problem becomes: the sum of two numbers is 21 and their difference is 3. What are the numbers? It can be solved with find two numbers by sum and difference calculator.

Add the 4th and 5th equations

(x + y) + (x – y) = 21 + 3

x + y + x – y = 24

2x = 24

x = 12

When x = 12, substitute it into any of the above equations to calculate the y value.

y = 9

So, the two numbers are 12 and 9.

Verify

12 * 9 = 108

12² + 9² = 144 + 81 = 225

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 product and sum of squares, 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.