Question

The difference of 2 natural numbers is 5 and the sum of their squares is 97. Which are the numbers.

Answer 1

Step-by-step explanation:

let the numbers be x and y

given that x - y = 5 and x^2 + y^2 = 97

$x - y = 5 \\ x = 5 + y \\ {x}^{2} + {y}^{2} = 97 \\ substitute \: x = 5 + y \: \\ {(5 + y)}^{2} + {y}^{2} = 97 \\ 25 + {y}^{2} + 10y + {y}^{2} = 97 \\ 2 {y}^{2} + 10y + 25 = 97 \\ 2 {y}^{2} + 10y + 25 - 97 = 0 \\ 2 {y}^{2} + 10y - 72 = 0 \\ 2( {y}^{2} + 5y - 36) = 0 \\ {y}^{2} + 5y - 36 = 0 \\ {y}^{2} + 9y - 4y - 36 = 0 \\ y(y + 9) - 4(y + 9) = 0 \\ (y + 9)(y - 4) = 0 \\ y + 9 = 0 \: \: \: \: y - 4 = 0 \\ y = - 9 \: \: \: \: \: y = 4$

we take the positive value of y = 4

x = 5 + y

x = 5 + 4

x = 9

therefore the numbers are 9 and 4

verification

x - y = 5

9 - 4 = 5

5 = 5

x^2 + y^2

= 9^2 + 4^2

= 81 + 16

= 97

hope you get your answer