Question

the difference of square of two numbers is 180 the square of smaller number is 8 times the larger number Find two numbers

Answer 1

Let the larger no be x
Then the square of the smaller be 8
8 times the larger no be 8x and the square of the larger no be x2
x2-8x=180
x2-8x-180=0
x2-18x+10x-180=0
x(x-18)+10(x-18)=0
(x-18) (x+10)=0
x=18 or x=-10
Thus the larger no be 18 or -10
Then the square of smaller no be 8(18) or 8(-10)
=144 or -80
The square cannot be negative
So the square no be 12

Answer 2

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

$\Large{\fbox{Assumption}}$

Smaller number = u

Larger number = v

★Condition

v² - u² = 180 ..... (1)

u² = 8v ..... (2)

★Put the value of u² in (1)

v² - 8v = 180

v² - 8v - 180 = 0

$\fbox{Splitting\;the\;middle\;term :-}$

v² - 18v + 10v - 180 = 0

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

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

(v - 18) = 0

v = 18

(v + 10) = 0

v = -10

$\text{Negative\;sign\;is\;rejecting}$

v = 18

Put the value in (2)

u² = 8v

u² = 8(18)

u² = 144

u = √144

u = 12

★Therefore :-

$\fbox{Two\;numbers\;are\;12\;and\;18}$