Expert: Scott A Wilson - 7/24/2009

Question
Hi,
Let x and y be positive integers such that x^2 + 3x +y^2 = 404. What is the value of x + y^3? Find all possible solutions.

How can I solve this using Excel (I have never used Excel for maths before, so I need some background information).

Thankyou very much,

Tim

Answer
First, lets give the columns titles.
In cell A1, put 404.  In Cell B1, put X.
In cell C1, put X^2+3X.  In cell D1, put Y.

Next, input the first row of values.
In cell A2, put 404.  In cell B2, put 1.  In cell C2, put =B1*(B1+3).
In cell D2, put =Sqrt(A3-C3).
The second row should now have, 404, 1, 4, and 20 in each of the columns.

Now the third row.
In cell A3, well, we’re not there yet.  Goto cell A2.  Do a control-C.
Now you can go to cell A3.  Do control-V.  That copied cell C2 to C3.
In cell B3, put =B2+1.  That makes B3 into a 2.

Here’s where it starts to get easy.
Goto C2.  Press F8 and hit the right arrow once.  
Cells C2 and D2 are highlighted.

Do a control-C again.  Goto cell C3.  Do a control-V.  

Those cells should now have
10   19.849433

In fact, the whole sheet should have
404   X   X^2+3X   Y
404   1   4   20
404   2   10   19.849433

Now goto cell A3.  Press F8 and the right arrow three times.
This should highlight cells A3 through D3, which to Excel is what’s known as A3:D3.
Goto A4.  Do a control-V.

Hit the down arrow and Goto A5.  Do a control-V
Now to get it all done at once, while the four cells A5:D5 are highlighted, hit F8.

Using the down arrow, make the highlight go down to row 15.  
You now have highlighted A5:D20.
Now use the mouse the click on Edit, Fill, Down.

And now you’ve got it.  That’s is all of the X values from 1 to 14.  
Row 2 has 1, 4, 20, so that is one of the even solutions.  
Row 13 has 14, 208, 14, and that is the another integer solution.  
Row 17 has 17, 340, and 8.  That is the 3rd integer solution.  
Row 18 has 17, 340, and 8.  That is the last integer solution.

The reason I know this one is that last is that in cell D20, there is a #NUM!  
This means that the spreadsheet is attempting to do what can’t be done,
which is taking the square-root of C20-A20.

Note that C20 is 418 and A20 is only 404.
That means it is trying to find =sqrt(-14).  That can’t be done.

Note that the cells with a many decimal Y indicate that there is a solution,
but that it can’t be found.  The data there is only a six decimal approximation to the answer.

The entire sheet should look like
404   X   X^2+3X   Y
404   1   4   20
404   2   10   19.849433
404   3   18   19.646883
404   4   28   19.390719
404   5   40   19.078784
404   6   54   18.708287
404   7   70   18.275667
404   8   88   17.776389
404   9   108   17.204651
404   10   130   16.552945
404   11   154   15.811388
404   12   180   14.96663
404   13   208   14
404   14   238   12.884099
404   15   270   11.575837
404   16   304   10
404   17   340   8
404   18   378   5.0990195
404   19   418   #NUM!

Now you may just read this, but actually it needs to be done in Excel so that you learn how to use it.  Excel is really a powerful piece of software.  It becomes even better the more you know about it.