SOLUTION: To the nearest ten minutes, find all values of x in the interval 0 less than or equal to x less than or equal to 360 that satisfy {{{ 3tan^2(x)+5tan(x)=2 }}}

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Question 754360: To the nearest ten minutes, find all values of x in the interval 0 less than or equal to x less than or equal to 360 that satisfy +3tan%5E2%28x%29%2B5tan%28x%29=2+
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
To the nearest ten minutes, find all values of x in the interval 0 less than or equal to x less than or equal to 360 that satisfy +3tan%5E2%28x%29%2B5tan%28x%29=2+
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Sub u for tan(x)
3u%5E2+%2B+5u+-+2+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B5x%2B-2+=+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=%285%29%5E2-4%2A3%2A-2=49.

Discriminant d=49 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-5%2B-sqrt%28+49+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%285%29%2Bsqrt%28+49+%29%29%2F2%5C3+=+0.333333333333333
x%5B2%5D+=+%28-%285%29-sqrt%28+49+%29%29%2F2%5C3+=+-2

Quadratic expression 3x%5E2%2B5x%2B-2 can be factored:
3x%5E2%2B5x%2B-2+=+%28x-0.333333333333333%29%2A%28x--2%29
Again, the answer is: 0.333333333333333, -2. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B5%2Ax%2B-2+%29

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tan(x) = 1/3
x = 18.435, 198.435 degs
tan(x) = -2
x = 116.565, 296.565 degs