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

2 Answers
Feb 8, 2016

Answer:

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

Feb 8, 2016

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#.