# The difference of two numbers is 3 and the difference of their square is 69. What are the numbers?

May 8, 2018

color(red)(10 and 13) or color(blue)(-10 and -13

#### Explanation:

Let $x \mathmr{and} y$ be the two numbers.$\left(x , y \in \mathbb{Z}\right)$

The difference of two numbers is 3.

$i . e . | x - y | = 3 \implies x - y = \pm 3$

=>color(red)(x-y=3...to(1) orcolor(blue)( x-y=-3...to(2)

The difference of their square is 69.

$i . e . {x}^{2} - {y}^{2} = 69$

$\implies \left(x - y\right) \left(x + y\right) = 69 ,$

where, color(red)(x-y=3 from (1)or color(blue)(x-y=-3 from $\left(2\right)$

$\implies 3 \left(x + y\right) = 69 \mathmr{and} - 3 \left(x + y\right) = 69$

=>color(red)(x+y=23...to(3)) or color(blue)( x+y=-23...to(4)

Adding $\textcolor{red}{\left(1\right) \mathmr{and} \left(3\right)} \mathmr{and} \textcolor{b l u e}{\left(2\right) \mathmr{and} \left(4\right)}$

color(red)(x-y=3)color(white)(............................... color(blue)( x-y=-3

color(red)(ul(x+y=23)color(white)(............................. color(blue)(ul(x+y=-23)
color(red)(2x=3+23=26)color(white)(................... color(blue)( color(blue)(2x=(-3)+(-23)=-26

color(red)(x=26/2)color(white)(.................................. color(blue)( x=-26/2

color(red)(x=13)color(white)(.................................... color(blue)(x=-13

Subst. color(red)(x=13 in equn. (3) or color(blue)(x=-13 in equn. $\left(4\right)$

we get

color(red)(13+y=23)color(white)(............................ color(blue)(-13+y=-23

=>color(red)(y=23-13)color(white)(....................... color(blue)(y=-23+13

=>color(red)(y=10)color(white)(..................................color(blue)(y=-10

Hence, the reqd. two numbers are:

color(red)(10 and 13) or color(blue)(-10 and -13