Question

the difference of squares of 2 numbers is 180. The squares of the smaller number is 8 times the larger number. Find the 2 number?

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Answer 1

$\Large{\textbf{\underline{\underline{According\:to\:the\:Question}}}}$

Assumption

${\boxed{\sf\:{Larger\;number\;be\;p}}}$

Hence,

${\boxed{\sf\:{Square\;of\;smaller\;Number}}}$

$\textbf{\underline{Eight\;times\;the\;Larger}}$

= 8p

${\boxed{\sf\:{Square\;of\;larger\;Number}}}$

= p

Situation,

p - 8p = 180

p - 8p - 180 = 0

p - 18p + 10p - 180 = 0

p(p - 18) + 10(p - 18) = 0

(p - 18) (p + 10) = 0

p - 18 = 0

p = 18

p + 10 = 0

p = -10

Therefore,

${\boxed{\sf\:{Larger\;Number}}}$

= 18 , -10

$\textbf{\underline{Negative\;is\;not\;applicable}}$

${\boxed{\sf\:{Square\;of\;smaller\;Number}}}$

= 8 × 18

= 144

8 × -10

= -80

$\textbf{\underline{Negative\;is\;not\;applicable}}$

${\boxed{\sf\:{Smaller\;Number}}}$

= √144

= 12

Hence

$\Large{\boxed{\sf\:{Numbers\;=\;12\;and\;18}}}$