The square of the sum of two consecutive integers is 1681. What are the integers?

Let's say the two consecutive numbers are #a# and #b#. We need to find an equation that we can solve to work out their values.

"The square of the sum of two consecutive integers is #1681#." That means if you add #a# and #b# together, then square the result, you get #1681#. As an equation we write:

#(a+b)^2=1681#

Now, there are two variables here so at first glance it looks unsolvable. But we're also told that #a# and #b# are consecutive, which means that #b=a+1#!

Substituting this new information in gives us:

#(a+a+1)^2=1681#
#(2a+1)^2=1681#

Next we're going to follow these steps to solve for #a#:

1) Take the square root of both sides. This will give two possible results, since both positive and negative numbers have positive squares.
2) Subtract #1# from both sides.
3) Divide both sides by #2#.
4) Check the answer.

#(2a+1)^2=1681#
#2a+1=sqrt(1681)=41#
#2a=40#
#a=20#

This means that #b=21#! To check these answers, take the values #20# and #21# and substitute them into the original equation like this:

#(a+b)^2=1681#
#(20+21)^2=1681#
#1681=1681#

Success!