# One number is 5 greater than another number.If the sum of their squares is 5 times the square of the smaller number,what are the numbers?

Jan 24, 2016

5 and 10

#### Explanation:

let the smaller number be n, then the 2nd will be n + 5

then : ${n}^{2} + {\left(n + 5\right)}^{2} = 5 {n}^{2}$

so ${n}^{2} + {n}^{2} + 10 n + 25 = 5 {n}^{2}$

and $3 {n}^{2} - 10 n - 25 = 0$

To factor : $3 \times \left(- 25\right) = - 75$

(require factors of (-75) that also sum to (-10) )

These are 5 and - 15 : now rewrite equation as

# 3n^2 - 15n + 5n - 25 = 0 and factoring gives

3n(n - 5 ) + 5 (n - 5 ) = 0 so (n - 5 )(3n + 5 ) = 0

$\Rightarrow n = 5 \mathmr{and} n = - \frac{5}{3}$

but n cannot be negative and hence n = 5 and n + 5 = 10