SOLUTION: The sum of the square of a positive number and the square of 3 more than the number is 65. What is the number?

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Question 945127: The sum of the square of a positive number and the square of 3 more than the number is 65. What is the number?
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
Let x be the positive number
x%5E2%2B%28x%2B3%29%5E2=65
x%5E2%2Bx%5E2%2B6x%2B9=65
2x%5E2%2B6x%2B9=65 Subtract 65 from each side
2x%5E2%2B6x-56=0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B6x%2B-56+=+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=%286%29%5E2-4%2A2%2A-56=484.

Discriminant d=484 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-6%2B-sqrt%28+484+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%286%29%2Bsqrt%28+484+%29%29%2F2%5C2+=+4
x%5B2%5D+=+%28-%286%29-sqrt%28+484+%29%29%2F2%5C2+=+-7

Quadratic expression 2x%5E2%2B6x%2B-56 can be factored:
2x%5E2%2B6x%2B-56+=+2%28x-4%29%2A%28x--7%29
Again, the answer is: 4, -7. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B6%2Ax%2B-56+%29

ANSWERS 4,-7 The number is restricted to positive numbers so the answer is 4.
CHECK:
x%5E2%2B%28x%2B3%29%5E2=65
4%5E2%2B%284%2B3%29%5E2=65
16%2B49=65
65=65