SOLUTION: two positive numbers differ by 11, and their square roots differ by 1. Find the numbers.

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: two positive numbers differ by 11, and their square roots differ by 1. Find the numbers.      Log On


   

Question 173445: two positive numbers differ by 11, and their square roots differ by 1. Find the numbers.
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
Let's go at this straight up and see what happens:

Let one of the numbers be x and the other one be y, so we know that:

x-y=11 and sqrt%28x%29-sqrt%28y%29=1.

Yuck! You can see right off that those radicals are going to make this messy.

So let's try another tack.

Let's keep x and y as the desired variables, but let's define two more variables, thus:

Let t=sqrt%28x%29 and u=sqrt%28y%29. That means that x=t%5E2 and y=u%5E2.

Now we can write:

1. t%5E2-u%5E2=11 and

2. t-u=1

Solve Eq. 2 for t, t=u%2B1 and substitute this expression for t in Eq. 1.

%28u%2B1%29%5E2-u%5E2=11

Expand the binomial and collect terms:

u%5E2%2B2u%2B1-u%5E2=11

2u%2B1=11

2u=10

u=5

Since we know that t-u=1, t=6

But x=t%5E2=36 and y=u%5E2=25

Hence, 36 and 25 are the two numbers.

Check:

36-25=11

sqrt%2836%29-sqrt%2825%29=6-5=1

Answer checks.