SOLUTION: the difference of nine more than 3 times a number and the square of one -third of the number is negative eight

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Question 940650: the difference of nine more than 3 times a number and the square of one -third of the number is negative eight
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
3n%2B9-%28n%2F3%29%5E2=-8
3n%2B9-n%5E2%2F9=-8
9%283n%2B9-n%5E2%2F9%29=-8%2A9 Multiply both sides by 9
27n%2B81-n%5E2=-72 Add 72 to each side
27n%2B81-n%5E2%2B72=-72%2B72
27n%2B153-n%5E2=0 Multiply by -1 and rearrange
n%5E2-27n-153=0 Solve the quadratic
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation an%5E2%2Bbn%2Bc=0 (in our case 1n%5E2%2B-27n%2B-153+=+0) has the following solutons:

n%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=%28-27%29%5E2-4%2A1%2A-153=1341.

Discriminant d=1341 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--27%2B-sqrt%28+1341+%29%29%2F2%5Ca.

n%5B1%5D+=+%28-%28-27%29%2Bsqrt%28+1341+%29%29%2F2%5C1+=+31.8098334236006
n%5B2%5D+=+%28-%28-27%29-sqrt%28+1341+%29%29%2F2%5C1+=+-4.80983342360055

Quadratic expression 1n%5E2%2B-27n%2B-153 can be factored:
1n%5E2%2B-27n%2B-153+=+1%28n-31.8098334236006%29%2A%28n--4.80983342360055%29
Again, the answer is: 31.8098334236006, -4.80983342360055. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-27%2Ax%2B-153+%29

(n-31.8)(n+4.8)=0
n=31.8, -4.8098
CHECK:
Solution 1:

(((114.4-112.36)=-8
2.04=-8 FALSE. Not a solution
Solution 2:
3n%2B9-%28n%2F3%29%5E2=-8
3%28-4.8098%29%2B9-%28-4.8098%2F3%29%5E2=-8
%28-14.429%2B9%29-%28-1.6%29%5E2=-8
-5.429-2.57=-8
-7.999=-8 This is the solution
The number is =-4.8098