Find 2 Numbers by Sum and Sum of Cubes Calculator

Welcome to the free online calculator to find two numbers by given sum and sum of cubes. For example, the sum of two numbers is 7 and the sum of their cubes is 133. 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 = 7………(1)

x³ + y³ = 133………(2)

Cube both sides of the first equation and simplify to get the third equation.

(x + y)³ = 7³

x³ + 3x²y + 3xy² + y³ = 343

x³ + y³ + 3xy(x + y) = 343………(3)

Substitute the first and second equations into the third equation to get the 4th equation,

133 + 3xy * 7 = 343

21xy = 210

xy = 10………(4)

Based on Equation 1 and Equation 4, the problem becomes that the sum of two numbers is 7 and their product is 10. What are the numbers? How to solve it? There are 2 methods, one is the quadratic equation method and the other is the factor method. For the specific process, please refer to the calculator finds two numbers by sum and product.

Here, the quadratic equation method is used to solve.

Since x + y= 7, so x = 7 – y. Put this value into the 4th equation.

(7 – y) y = 10

7y – y² = 10

y² – 7y + 10 = 0

(y – 2) (y – 5) = 0

y = 2 or y = 5

From the relationship between x and y, we can easily calculate the value of x.

x = 5 or x = 2

So, the two numbers are 2 and 5.

Let’s verify.

2 + 5 = 7

2³ + 5³ = 8 + 125 = 133

Correct.

Obviously, it is not simple to find two numbers based on the sum and the sum of cubes by hand. So, we developed this free online calculator to find two numbers by the sum and sum of cubes. The calculator omits all steps and displays the answer in milliseconds.

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.