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.
Answers
Answered by
162
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
⇒ 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
Answered by
205
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.
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