SOLUTION: solve he equation {{{(2x-1)^2=x^2-2}}}

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Question 156376: solve he equation %282x-1%29%5E2=x%5E2-2
Found 2 solutions by Alan3354, jim_thompson5910:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
%282x-1%29%5E2=x%5E2-2
4x%5E2+-+4x+%2B+1+=+x%5E2-2
3x%5E2+-+4x+%2B+3+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B-4x%2B3+=+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=%28-4%29%5E2-4%2A3%2A3=-20.

The discriminant -20 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 -20 is + or - sqrt%28+20%29+=+4.47213595499958.

The solution is , or
Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B-4%2Ax%2B3+%29

The onsite solver doesn't make it clear.
It's (2 +/- sqrt(5)i)/3 where i = sqrt(-1)



Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
%282x-1%29%5E2=x%5E2-2 Start with the given equation


4x%5E2-4x%2B1=x%5E2-2 FOIL the left side


4x%5E2-4x%2B1-x%5E2%2B2=0 Subtract x%5E2 from both sides. Add 2 to both sides.


3x%5E2-4x%2B3=0 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=3, b=-4, and c=3


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28-4%29+%2B-+sqrt%28+%28-4%29%5E2-4%283%29%283%29+%29%29%2F%282%283%29%29 Plug in a=3, b=-4, and c=3


x+=+%284+%2B-+sqrt%28+%28-4%29%5E2-4%283%29%283%29+%29%29%2F%282%283%29%29 Negate -4 to get 4.


x+=+%284+%2B-+sqrt%28+16-4%283%29%283%29+%29%29%2F%282%283%29%29 Square -4 to get 16.


x+=+%284+%2B-+sqrt%28+16-36+%29%29%2F%282%283%29%29 Multiply 4%283%29%283%29 to get 36


x+=+%284+%2B-+sqrt%28+-20+%29%29%2F%282%283%29%29 Subtract 36 from 16 to get -20


x+=+%284+%2B-+sqrt%28+-20+%29%29%2F%286%29 Multiply 2 and 3 to get 6.


x+=+%284+%2B-+2i%2Asqrt%285%29%29%2F%286%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%284%2B2i%2Asqrt%285%29%29%2F%286%29 or x+=+%284-2i%2Asqrt%285%29%29%2F%286%29 Break up the expression.


x+=+%282%2Bi%2Asqrt%285%29%29%2F%283%29 or x+=+%282-i%2Asqrt%285%29%29%2F%283%29 Reduce


So the answers are x+=+%282%2Bi%2Asqrt%285%29%29%2F%283%29 or x+=+%282-i%2Asqrt%285%29%29%2F%283%29


which approximate to x=0.667%2B0.745%2Ai or x=0.667-0.745%2Ai