which square numbers have a difference of 19?

Question

I need this help for my homework

Answer 1

Hi Magarchi!

If the question is asking about consecutive perfect squares, then you can setup a system of equations. Let's say x and y are two consecutive integers (i.e. their difference = 1) and x^2 is 19 more than y^2.

x - y = 1

x^2 - y^2 = 19

To solve this system of equations, we can rearrange the first equation to solve for x, substitute the result into the second equation, determine the value of y, and plug that back into the first equation to determine the value of x. (You can also reverse the order and determine the value of x followed by y.)

x = 1 + y

(1+y)^2 - y^2 = 19

(1 + 2y + y^2) - y^2 = 19

1 + 2y = 19

2y = 18

y = 9

Thus, our smaller integer (i.e. y) is 9. Since x = 1 + y, the larger integer (i.e. x) is 10. The two squares are 81 and 100, and their difference is 19.

-Kaleab