SOLUTION: sqrt(2x+3) + sq.rt.(x-2) = 2

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Question 46100: sqrt(2x+3) + sq.rt.(x-2) = 2
Answer by Nate(3500) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%282x+%2B+3%29+%2B+sqrt%28x+-+2%29+=+2
sqrt%282x+%2B+3%29+=+2+-+sqrt%28x+-+2%29
2x+%2B+3+=+%282+-+sqrt%28x+-+2%29%29%5E2
2x+%2B+3+=+2+-+4%2Asqrt%28x+-+2%29+%2B+x+-+2
x+%2B+3+=+-4%2Asqrt%28x+-+2%29
-x%2F4+-+3%2F4+=+sqrt%28x+-+2%29
%28-x%2F4+-+3%2F4%29%5E2+=+x+-+2
x%5E2%2F16+%2B+3x%2F16+%2B+3x%2F16+%2B+9%2F16+=+x+-+2
x%5E2%2F16+%2B+6x%2F16+%2B+9%2F16+=+x+-+2
x%5E2+%2B+6x+%2B+9+=+16x+-+32
x%5E2+-+10x+%2B+41+=+0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-10x%2B41+=+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=%28-10%29%5E2-4%2A1%2A41=-64.

The discriminant -64 is less than zero. That means that there are no solutions among real numbers.

If you are a student of advanced school algebra and are aware about imaginary numbers, read on.


In the field of imaginary numbers, the square root of -64 is + or - sqrt%28+64%29+=+8.

The solution is x%5B12%5D+=+%28--10%2B-+i%2Asqrt%28+-64+%29%29%2F2%5C1+=++%28--10%2B-+i%2A8%29%2F2%5C1+

Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-10%2Ax%2B41+%29

graph%28600%2C600%2C-10%2C10%2C-10%2C10%2Csqrt%282x+%2B+3%29+%2B+sqrt%28x+-+2%29%2C2%29
There are no solutions.