SOLUTION: Find two integers whose sum is 96 and whose product is 1728.

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Question 51433: Find two integers whose sum is 96 and whose product is 1728.
Answer by rchill(405) About Me  (Show Source):
You can put this solution on YOUR website!
Let x and y represent the two integers. We now have equations x%2By=96 and xy=1728. Using the second equation, let's solve for y by dividing both sides by x: %28xy%29%2Fx=1728%2Fx, which simplifies to y=1728%2Fx. Now use this value of y and substitute it into the first equation thusly: x%2B1728%2Fx=96. Multiplying both sides by x produces x%5E2%2B1728=96x. Now subtract 96x from both sides to yield x%5E2-96x%2B1728=0. Now we just have to factor this equation into (x - something) and (x - something_else), where the 'something' and 'something_else' when multiplied together produces 1728, and when added together they produce -96.

The best way to do this is not by trial and error, but by using the discriminant.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-96x%2B1728+=+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=%28-96%29%5E2-4%2A1%2A1728=2304.

Discriminant d=2304 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--96%2B-sqrt%28+2304+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-96%29%2Bsqrt%28+2304+%29%29%2F2%5C1+=+72
x%5B2%5D+=+%28-%28-96%29-sqrt%28+2304+%29%29%2F2%5C1+=+24

Quadratic expression 1x%5E2%2B-96x%2B1728 can be factored:
1x%5E2%2B-96x%2B1728+=+1%28x-72%29%2A%28x-24%29
Again, the answer is: 72, 24. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-96%2Ax%2B1728+%29