SOLUTION: Please show how to solve the following polynomial equations: {{{ x^4 - 2x^2 + 1 = 0 }}} by subbing y= x^2 and {{{ sqrt(x) - (8) root(4,x) + 7 }}} Thank you

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Please show how to solve the following polynomial equations: {{{ x^4 - 2x^2 + 1 = 0 }}} by subbing y= x^2 and {{{ sqrt(x) - (8) root(4,x) + 7 }}} Thank you      Log On


   

Question 1018485: Please show how to solve the following polynomial equations:
+x%5E4+-+2x%5E2+%2B+1+=+0+ by subbing y= x^2
and
+sqrt%28x%29+-+%288%29+root%284%2Cx%29+%2B+7+

Thank you
Answer by ikleyn(52832) About Me  (Show Source):
You can put this solution on YOUR website!
.
Please show how to solve the following polynomial equations:
+x%5E4+-+2x%5E2+%2B+1+=+0+ by subbing y= x^2
and
+sqrt%28x%29+-+%288%29+root%284%2Cx%29+%2B+7+

Thank you
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1.
x%5E4+-+2x%5E2+%2B+1 = 0.  --->

Factor left side:

%28x%5E2-1%29%5E2 = 0,  --->

%28x-1%29%5E2%2A%28x%2B1%29%5E2 = 0,  --->  

The roots are x = 1 of multiplicity 2  and  x = -1 of multiplicity 2.

So it is solved.

Another way is to introduce new variable y = x%5E2. The you get your equation for y in the form

y%5E2+-+2y+%2B+1 = 0.

Again, factor left side

%28y-1%29%5E2 = 0,

and so on . . . You will get the same answer.



2. Regarding your second equation . . . 

   There is a mistake in your writing, so fix (repair) it, and then resubmit.