SOLUTION: If g(x) = √( x - 1) + √(2x), find all values of x for which g(x) = 6

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Question 245970: If g(x) = √( x - 1) + √(2x), find all values of x for which g(x) = 6
Answer by edjones(8007) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt(x-1)+sqrt(2x)=6
sqrt(x-1)=6-sqrt(2x)
x-1=2x-12sqrt(2x)+36
-x-37=-12sqrt(2x)
x^2+74x+1369=288x square each side
x^2-214x+1369=0
x=6.60... The other answer is extraneous.
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Ed
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Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-214x%2B1369+=+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=%28-214%29%5E2-4%2A1%2A1369=40320.

Discriminant d=40320 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--214%2B-sqrt%28+40320+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-214%29%2Bsqrt%28+40320+%29%29%2F2%5C1+=+207.399203184089
x%5B2%5D+=+%28-%28-214%29-sqrt%28+40320+%29%29%2F2%5C1+=+6.60079681591094

Quadratic expression 1x%5E2%2B-214x%2B1369 can be factored:
1x%5E2%2B-214x%2B1369+=+1%28x-207.399203184089%29%2A%28x-6.60079681591094%29
Again, the answer is: 207.399203184089, 6.60079681591094. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-214%2Ax%2B1369+%29