SOLUTION: The digit of a two-digit number adds up to 14. the number itself is 11 less than the sum of the squares of the digits. find the number Note: I worked out that the two numb

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Question 689851: The digit of a two-digit number adds up to 14. the number itself is 11 less than the sum of the squares of the digits. find the number



Note: I worked out that the two numbers are 5 and 9 by random guessing and substituting the numbers into the question. but I don't know the correct way to set out the working.
Answer by vleith(2983) About Me  (Show Source):
You can put this solution on YOUR website!
Let the tens digit be t and the ones digit be 1 .
You are told t+%2B+o+=+14 --> t+=+14+-+o
Also 10t+%2B+o+=+t%5E2+%2B+o%5E2++-+11
So
10%2A%2814+-+o%29+%2B+o+=+%2814-o%29%5E2+%2B+o%5E2+-+11
140+-+9o+=+196+-+28o+%2B+o%5E2+%2B+o%5E2+-+11
0+=+2o%5E2+-+19o+%2B+45
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B-19x%2B45+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-19%29%5E2-4%2A2%2A45=1.

Discriminant d=1 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--19%2B-sqrt%28+1+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-19%29%2Bsqrt%28+1+%29%29%2F2%5C2+=+5
x%5B2%5D+=+%28-%28-19%29-sqrt%28+1+%29%29%2F2%5C2+=+4.5

Quadratic expression 2x%5E2%2B-19x%2B45 can be factored:
2x%5E2%2B-19x%2B45+=+2%28x-5%29%2A%28x-4.5%29
Again, the answer is: 5, 4.5. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-19%2Ax%2B45+%29

So the only integer answer is o = 5. Which means t = 14-5 = 9