Find Two Numbers by Sum and Squares Calculator

Welcome to the free online calculator to find two numbers by sum and sum of squares. For example, the sum of two numbers is 17 and the sum of their squares is 145. 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 = 17………(1)

x² + y² = 145………(2)

Simplify the first equation, let x be represented by y.

x = 17 – y

Substitute x = 17 - y into the second equation and simplify.

x² + y² = 145

(17 – y)² + y² = 145

289 – 34y + y² + y² = 145

2y² – 34y + 289 = 145

2y² – 34y + 144 = 0

y² – 17y + 72 = 0

Now, we get a quadratic equation, the next key is to solve this quadratic equation. The answer to the quadratic equation can be calculated using the formula method or the factorization method. The quadratic formula is

Among them, a = 1, b = -17, c = 72. So, we can get the value of y

b² – 4ac = (-17)² – 4 * 1 * 72 = 289 – 288 = 1

y = -(-17) + √12*1 = 17 + 12 = 9

Or

y = -(-17) – √12*1 = 17 – 12*1 = 8

Or use the factorization method directly.

y² – 17y + 72 = 0

(y – 8) (y – 9) = 0

y = 8 or y = 9

Both methods can calculate the value of y.

When y = 8, according to the relationship between x and y, we can calculate the value of x.

x = 17 – y = 17 – 8 = 9

When y = 9, the same operation can be used to calculate x = 8.

Therefore, the two numbers are 8 and 9.

Is it right or not? Check it out!

8 + 9 = 17

8² + 9² = 64 + 81 = 145

Yes, this is correct.

Of course, if you don’t want to calculate quadratic equations, we can think differently.

Square both sides of the first equation and simplify.

x + y = 17

(x + y)² = 17²

x² + 2xy + y² = 289

Then subtract the second equation from this new equation.

(x² + 2xy + y²) – (x² + y²) = 289 – 145

x² + 2xy + y² – x² – y² = 144

2xy = 144

xy = 72

Did you find anything?

Yes, x + y = 17, xy = 72. The question becomes, the sum of two numbers is 17 and their product is 72. What are the numbers? For specific solutions, see Find Two Numbers by Sum and Product Calculator.

In conclusion, solving such problems is not difficult, but requires simple and tedious calculations. So, we developed this free online calculator to find two numbers by sum and sum of squares. 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.