SOLUTION: Solve the following equation, giving the exact solutions which lie in [0, 2π). (Enter your answers as a comma-separated list.) 2 tan(x) = 1 − tan^2(x)

Algebra ->  Finance -> SOLUTION: Solve the following equation, giving the exact solutions which lie in [0, 2π). (Enter your answers as a comma-separated list.) 2 tan(x) = 1 − tan^2(x)      Log On


   

Question 1078822: Solve the following equation, giving the exact solutions which lie in [0, 2π). (Enter your answers as a comma-separated list.)
2 tan(x) = 1 − tan^2(x)
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
2 tan(x) = 1 - tan^2(x)
tan^2 + 2tan - 1 = 0
Sub x for tangent
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Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B2x%2B-1+=+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=%282%29%5E2-4%2A1%2A-1=8.

Discriminant d=8 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-2%2B-sqrt%28+8+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%282%29%2Bsqrt%28+8+%29%29%2F2%5C1+=+0.414213562373095
x%5B2%5D+=+%28-%282%29-sqrt%28+8+%29%29%2F2%5C1+=+-2.41421356237309

Quadratic expression 1x%5E2%2B2x%2B-1 can be factored:
1x%5E2%2B2x%2B-1+=+%28x-0.414213562373095%29%2A%28x--2.41421356237309%29
Again, the answer is: 0.414213562373095, -2.41421356237309. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B2%2Ax%2B-1+%29

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tan(x) = sqrt(2) - 1
tan(x) = -sqrt(2) - 1
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Find x with a calculator