Question

the sum of the squares of two positive integer is 208. if the square of the larger number is 18 times the smaller number find the numbers.

Answer 1

GIVEN ;-

⇒ Sum of Squares of two positive numbers are  = 208

⇒Sum of larger number is 18 times  the smaller number

TO FIND ;-

⇒ Find the two numbers = ?

SOL ;-

⇒ Let us take one of the number as x ,

⇒ Also the other number as y .
 
According to the  question , { ''  if the square of the larger number is 18 times the smaller number " }

So by this statement we can write as ,

   ⇒           x²  =  18 y
 

⇒ x²  +  y²  = 208 { because the sum of the squares of two positive integer is                                 }208.

⇒  x²  +  y²  = 208

⇒ Let us find the y 

⇒18  y  +  y²   =   208

⇒   y²   +  18  y  -   208   =  0

⇒   y²   +  26  y -  8y  -    208   =  0

⇒  y  (  y + 26 ) - 8 ( y + 26 )   =   0

⇒  (  y+ 26  )  (  y  -  8  )         = 0

But in the question it has been given that the integers are in the positive.

⇒  (  y  -  8  )         =  0   

⇒         y               = 8

We got the value of one integer , now let us find the value of the another integer.......
 

x²  +  y²  = 208

x²  +  ( 8 )²  =  208

x²  +  64      =208

         x²       =  208-64

         x²       =  144
        
         x        =√ 144
        
         x         = 12

So the two numbers are - 12 and 8

Answer 2

Answer:

Step-by-step explanation:

Solution :-

Let the smaller number be x,

Then,

Square of the larger number = 18x

Square of the smaller number = x²

Sum of the squares of the integers = 208

According to the question,

⇒ x² + 18x = 208

⇒ x² + 18x - 208 = 0

By using prime factorization method, we get

⇒ x² + 26x - 8x - 208 = 0

⇒ x(x + 26) - 8(x + 26) = 0

⇒ (x + 26) (x - 8) = 0

⇒ x + 26 = 0 or x - 8 = 0

⇒ x = - 26, 8 (As x can't be negative)

⇒ x = 8

Square of larger number = 18x = 18 × 8 = 144

Larger number = √144 = 12

Hence, the number are 8 and 12.