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

Feb 8, 2016

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 : ${n}^{2} + {\left(n + 2\right)}^{2} = 202$

expanding gives : ${n}^{2} + {n}^{2} + 4 n + 4 = 202$

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

$2 {n}^{2} + 4 n - 198 = 0$

common factor of 2 : $2 \left({n}^{2} + 2 n - 99\right) = 0$

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

Feb 8, 2016

Always remember that color(blue)(odd color(blue)(ncolor(blue)(umbers always differ in the value of $\textcolor{g r e e n}{2}$

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.