# The sum of two numbers is 18 and the sum of their squares is 170. How do you find the numbers?

Mar 5, 2016

7 and 11

#### Explanation:

a) x+y=18

b) x^2+y^2=170

a) y=18-x

replace y in b)

b) x^2+(18-x)^2=170

${x}^{2} + 324 - 36 x + {x}^{2} = 170$

$2 {x}^{2} - 36 x + 324 - 170 = 0$

$2 {x}^{2} - 36 x + 154 = 0$

Now you only need to use the quadratic form:

$x = \frac{36 \pm \sqrt{{36}^{2} - 4 \cdot 2 \cdot 154}}{2 \cdot 2}$

$x = \frac{36 \pm \sqrt{1296 - 1232}}{4}$

$x = \frac{36 \pm \sqrt{64}}{4} = \frac{36 \pm 8}{4}$

$x = \frac{36 + 8}{4} \mathmr{and} x = \frac{36 - 8}{4}$

$x = 11 \mathmr{and} x = 7$ and $y = 18 - 11 = 7 \mathmr{and} y = 18 - 7 = 11$

So, the numbers are 7 and 11