Question

The sum of the square of two consecutive positive odd integers is 202, how do you find the integers?

Answer 1

9 , 11

Explanation:

let n be a positive odd integer

then the next consecutive odd number is , n + 2 , since odd numbers have a difference of 2 between them.

from the given statement : `n^2 + (n+2)^2 = 202`

expanding gives : `n^2 + n^2 + 4n + 4 =202`

this is a quadratic equation so collect terms and equate to zero.

`2n^2 + 4n -198 = 0`

common factor of 2 : `2(n^2 + 2n - 99) = 0`

now consider factors of -99 which sum to +2. These are 11 and -9.

hence : 2(n + 11 )(n-9 ) = 0

(n + 11 ) = 0 or (n-9) = 0 which leads to n = -11 or n = 9

but n > 0 hence n = 9 and n+ 2 = 11

Answer 2

Always remember that `color(blue)(odd` `color(blue)(n` `color(blue)(umbers` always differ in the value of `color(green)2`

So,let the first number be `color(red)(x`

Then the second number will be `color(red)(x+2`

Then,

`color(green)((x)^2+(x+2)^2=202`

Use formula `color(green)((a+b)^2color(blue)(=color(brown)(a^2+2ab+b^2`

`rarrcolor(green)(x^2+x^2+2x(2)+2^2=color(blue)(202`

`rarrcolor(green)(x^2+x^2+4x+4=color(blue)(202`

`rarrcolor(green)( 2x^2+4x+4=color(blue)(202`

`rarrcolor(green)( 2x^2+4x=color(blue)(202-4`

`rarr color(green)(2x^2+4x=color(blue)(198`

`rarrcolor(green)( 2x^2+4x-198=color(blue)(0`

Now this is a Quadratic equation (in form `color(brown)(ax^2+bx^2+c=0`) So,we can use the Quadratic formula or factor it out.

Luckily,we can factor it to

`rarrcolor(green)( 2x^2+4x-198=color(brown)((2x+22)(a-9)=0`

`rarrcolor(green)( (2x+22)(a-9)=color(brown)(0`

Now we have two values for `color(green)(x` which are

`1)rarr color(green)(x=-22/2=-11`

`2)rarrcolor(green)( x=9`

Now we need to find `color(orange)(x+2`

If `color(green)(x=-11`

Then,`color(orange)(x+2=-11+2=-9`

And if `color(green)(x=9`

Then,`color(orange)(x+2=9+2=11`

So,at the end we conclude the if the first integer is `color(green)(-11` then second integer is `color(orange)(-9` and if the first integer is `color(green)(9`,the second integer is `color(orange)(11`.