SOLUTION: Find two integers, x and x + 1, whose squares differ by 25.

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Question 117950: Find two integers, x and x + 1, whose squares differ by 25.
Found 3 solutions by MathLover1, bucky, jim_thompson5910:
Answer by MathLover1(20849) About Me  (Show Source):
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two integers;
x and x+%2B+1, whose squares differ by 25
x%5E2+%2B+25+=+%28x+%2B+1%29%5E2
x%5E2+%2B+25+=+x%5E2+%2B+2x+%2B+1

x%5E2+-+x%5E2+=++2x+%2B+1+-+25+
0+=++2x+-+24
0+=++x+-+12
x=++12

and x+%2B+1=+13

12%5E2=144
13%5E2=169
169-144=25

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
The two unknown integers are represented by x and by x+1
.
If you square the two integers you get:
.
x%5E2
and
%28x%2B1%29%5E2
.
Squaring %28x%2B1%29 results in %28x%2B1%29%5E2+=+%28x%2B1%29%2A%28x%2B1%29+=+x%5E2+%2B+2x+%2B+1
.
If you subtract the two squares, the problem tells you that the result is 25. In equation form
this subtraction can be expressed as:
.
x%5E2+%2B+2x+%2B1+-+x%5E2+=+25
.
Notice that in the subtraction the two x%5E2 terms are of opposite sign and therefore
cancel each other. Therefore, you are left with:
.
2x+%2B+1+=+25
.
Get rid of the 1 on the left side by subtracting 1 from both sides to get:
.
2x+=+24
.
Solve for x by dividing both sides of this equation by 2 to reduce the equation to:
.
x+=+24%2F2+=+12
.
So one of the unknown integers (that is x) equals 12. The other integer is x + 1 and therefore
it equals 12 + 1 or 13.
.
The two integers you are looking for are 12 and 13.
.
Check. 12 squared is 144 and 13 squared is 169. The difference is 169 - 144 = 25, just as the
problem requires. So this answer checks and the two integers are 12 and 13.
.
Hope this helps you to understand the problem.
.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If their squares differ by 25, then we have the equation %28x%2B1%29%5E2-x%5E2=25


%28x%2B1%29%5E2-x%5E2=25 Start with the given equation


x%5E2%2B2x%2B1-x%5E2=25 Foil


2x%2B1=25 Combine like terms



2x=25-1Subtract 1 from both sides


2x=24 Combine like terms on the right side


x=%2824%29%2F%282%29 Divide both sides by 2 to isolate x



x=12 Divide

--------------------------------------------------------------
Answer:
So our answer is x=12



Now add 1 to 12 to get the next number. So x%2B1=12%2B1=13


This means our two numbers are 12 and 13


Check:

%28x%2B1%29%5E2-x%5E2=25 Start with the given equation

13%5E2-12%5E2=25 Plug in the solution

169-144=25 Square each term

25=25 Subtract. Since the two sides of the equation are equal, this verifies our answer.