Question

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

Answer 1

let the larger number be 'x'

and the smaller number be 'y'

ATQ,

$x { }^{2} - y {}^{2} = 180$

let it be equation 1

also,

$y {}^{2} = 8x$

now evaluating it in equation 1

$x {}^{2} - 8x - 180 = 0$

as now by solving the quadratic equation the answers are 18,-10

so we need to take the positive one only so the value of x is 18 and y is 12

Answer 2

Let the larger number be x and smaller number be y.
So, x^2- y^2 = 180
y^2 = x^2 - 180 ..... Equation 1


ATQ,
y^2 = 8x
Putting the value of y^2 from equation 1, we get:
x^2 - 180= 8x
x^2 -8x - 180 = 0
x^2 - (18-10)x - 180=0
x^2 - 18x + 10x - 180 = 0
x(x-18) +10(x-18) = 0
Either,x= - 10
Or,x=18
When x=10,
y= root( - 80)

When x = 18
y = 12....