The sum of the squares of two consecutive odd numbers is 394. Find the numbers.
Answers
Answer:
given: The sum of the squares of two consecutive odd numbers is 394.
To find: the numbers.
Solution:
Let the consecutive odd number be ‘a’ and a + 2According to given condition,
a2 + (a + 2)2 = 394
Use the formula (x+y)2=x2+y2+2xy in (a + 2)2
Here x=a and y=2,
⇒ a2 + a2 + 4 + 4a=394
⇒ 2a2 + 4a +4 – 394 = 0
⇒ 2a2 + 4a – 390 = 0
Take 2 common out of the above equation,
⇒ a2 + 2a – 195 = 0
Factorise by splitting the middle term.
⇒ a2 + 15a – 13a – 195 = 0
⇒ a(a + 15) – 13(a + 15) = 0
⇒ (a – 13)(a + 15) = 0
Thus, a = 13, - 15
When a=13 then a+2=15And when a = -15 then a+2 = -13
So Consecutive odd numbers are 13, 15 and -15,-13.
Answer :-
Consecutive odd nunmbers are - 13, - 15 or 13, 15.
Explanation :-
Let the two consecutove odd numbers be x, (x + 1)
Square of x = (x)² = x²
Square of (x + 2) = (x + 2)²
Given
Sum of the squares of two consecutive odd numbers = 394
⇒ x² + (x + 2)² = 394
⇒ x² + (x)² + 2(x)(2) + (2)² = 394
[ ∵ (a + b)² = a² + 2ab + b² ]
⇒ x² + x² + 4x + 4 = 394
⇒ 2x² + 4x + 4 = 394
⇒ 2(x² + 2x + 2) = 394
⇒ x² + 2x + 2 = 394/2
⇒ x² + 2x + 2 = 197
⇒ x² + 2x + 2 - 197 = 0
⇒ x² + 2x - 195 = 0
By splitting the middle term
⇒ x² + 15 - 13x - 195 = 0
⇒ x(x + 15) - 13(x + 15) = 0
⇒ (x + 15)(x - 13) = 0
⇒ x + 15 = 0 or x - 13 = 0
⇒ x = - 15 or x = 13
When x = - 15
One of the odd number = x = - 15
Another odd number = (x + 2) = (-15 + 2) = - 13
When x = 13
One of the odd number = x = 13
Another odd number = (x + 2) = (13 + 2) = 15
∴ the consecutive odd nunmbers are - 13, - 15 or 13, 15.