find two consecutive whole niumbers such that the sum of their squares is 265?

Answer 1

Let the first number be x
the consecutive number will be x+1

x^2 + (x+1)^2 = 265
x^2 + x^2 + 1 + 2x = 265
2x^2 + 2x - 264 = 0
x^2 + x - 132 = 0
x = [-1 +- sqrt( 1 + 4*132) ] / 2
x = (-1 +- 23) / 2
x = -24/2 and 22/2
ignore negative value of x as whole number doesnt include negative integers,
thus
x = 22/2
or x = 11
so the 2 numbers are
11 and 12

Answer 2

1) Each number must be less than [ √265 ] = 16.
2) One number is odd and the other is even as 265 is odd.
3) Subtract squares of all odd numbers less than 16 from 265 and look for the perfect square.

265 - (1)^2 = 264, 265 - (3)^2 = 256,
265 - (5)^2 = 240, 265 - (7)^2 = 216,
265 - (9)^2 = 184, 265 - (11)^2 = 144,
265 - (13)^2 = 96, 265 - (15)^2 = 40.

4) of the above only 256 = (16)^2 and 144 = (12)^2 are perfect squares.

So, answer is 3 and 16 or 11 and 12.
Since consecutive whole numbers are asked, it is 11 and 12.

After posting the answer, I saw other answers and feel that the algebraic method is superior. The reason is that it will work better with larger sum, whereas arithmetical method used by me as above will be tedious.

Answer 3

x =1st no., x + 1 = 2nd no.

Formula:
x^2 + (x + 1)^2 = 265
x^2 + x^2 + x + x + 1 = 265
2x^2 + 2x = 264
x^2 + x = 132
x^2 + x - 132 = 0
(x + 12)(x - 11) = 0

Answer: whole numbers are 11 and 12 or - 12 and - 11

Proof(265 is the total):
= 11^2 + 12^2
= 121 + 144
= 265

Answer 4

11 and 12

Answer 5

11 and 12

Answer 6

let x, x+1 be the consecutive whole numbers
x^2 + (x+1)^2 = 265
x^2 + x^2+2x+1 =265
2x^2+2x-264=0 --- (1)
you'll have to solve this quadratic equation
ax^2+bx+c=0
x=[-b+-sqrt(b^2-4ac)]/2a
a=2 b=2 c=-264
x=11 and -12
The consecutive numbers are either 11,12
or -12,-11
Note: you can divide 2x^2+2x-264=0 by 2
it becomes x^2+x-132=0 and factor it out as
(x+12)(x-11)=0 if you are good at numbers.
You'll get the same answer.

Answer 7

x² + (x + 1)² = 265
x² + x² + 2x + 1 = 265
2x² + 2x - 264 = 0
x² + x - 132 = 0
(x + 12)(x - 11) = 0
x = 11
11 and 12 is answer.

Answer 8

"Two consecutive whole numbers" suggests you should start with n and n+1.

Then you have n^2 + (n+1)^2 = 256

Just rearrange and solve the quadratic.