Question

the sum of squares of two consecutive odd numbers is 394.find the numbers.

Answer 1

Topic

Quadratic Equation

Given

Sum of squares of two consecutive odd numbers is 394.

To Find

Numbers satisfying given statement.

Solving

Let first number be x.

Next odd number will be ( x + 2 ).

x² + ( x + 2 )² = 394

x² + x² + 4x + 4 = 394

2x² + 4x = 390

x² + 2x = 195

x² + 2x - 195 = 0

x² + 15x - 13x -195 = 0

x( x + 15 ) - 13( x + 15 ) = 0

( x + 15 )( x - 13 ) = 0

Now, we will make cases

Case 1

x + 15 = 0

x = -15

then other number will be

x + 2 = -15 + 2 = -13

Case 2

x - 13 = 0

x = 13

then other number will be

x + 2 = 13 + 2 = 15

Answer

So, answer is :

-13 and -15

13 and 15

Note :-

Negative numbers are also odd numbers.

Verification

For Case 1

( -13 )² + ( -15 )²

169 + 225

394

For Case 2

( 13 )² + ( 15 )²

169 + 225

394

Hence, verified.

Answer 2

let one number be (2n-1), then the next consecutive odd number will be (2n+1)

(2n-1)^2 + (2n+1)^2 = 394
4n^2 +1 -4n +4n^2 +1+4n = 394
8n^2 +2 = 394
8n^2 = 392
n^2 = 49
n = 7

The two consecutive numbers are :
(2n-1) = 14-1 = 13
(2n+1) = 14+1= 15

your answer is 13 and 15.

hope this helped...