Question

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

Answer 1

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Lets suppose the two numbers are X and Y, where Y is the smaller number of the two, then
x^2 - y^2 = 180-----------1 
the square of the smaller number is eight times the larger number, so
y^2 = 8x--------------------2 
So, using equation 2 in equation 1, we have
x^2 - 8x = 180
x^2 - 8x - 180 = 0
splitting -8x into 10x - 18x
x^2 +10x - 18x - 180 = 0
x(x+10) -18(x+10) = 0
(x - 18)(x +10) = 0 
so x = 18 or x = -10 
If x = 18, using equation 2,
y^2 = 8 X 18 = 144
then y = +-12 
If x = -10, using equation 2,
y^2 = 8 X -10 = -80
then y will be an imaginary number. 
considering you are using only real numbers, the solution will be x = 18 and y = +-12. 

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Answer 2

$\large \sf \: We \: have,$

• Let the large number is = x

• Square of smaller number is = 8x

Now according to the given question, ㅤㅤㅤ

$\small\sf \green ⇢\: x^{2} - 8x = 180$

$\small \sf \green⇢ x^{2} - 8x - 180 = 0$

$\small \sf \green ⇢x^{2} - (18 - 10)x - 180 = 0$

$\small \sf \green ⇢x^{2} - 18x + 10x - 180 = 0$

$\small \sf \red ⇢x \: (x - 18) + 10 \: (x - 18) = 0$

$\small \sf \red ⇢(x - 18) \: (x + 10) = 0$

$\small \sf \red ⇢x - 18 = 0, \: x + 10 = 0$

$\small \sf \red ⇢x = 18, \: x = - 10$

Either

$\small \sf x=-10 \: and \: x=18$

$\small \sf x=18 \: is \: true \: (Positive \: Value)$

Now, square of small number

$\small \sf =18x =18 \times 8=144$

$\sf =\sqrt{144}$

$\small \sf = 12$

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