SOLUTION: I do not know how to factor this equation: x^2-2xcos(theta)+1=0 in terms of x What I am looking for are the roots

Algebra ->  Equations -> SOLUTION: I do not know how to factor this equation: x^2-2xcos(theta)+1=0 in terms of x What I am looking for are the roots      Log On


   

Question 443789: I do not know how to factor this equation:
x^2-2xcos(theta)+1=0 in terms of x
What I am looking for are the roots
Answer by swincher4391(1107) About Me  (Show Source):
You can put this solution on YOUR website!
What you need to figure out is what multiplies to 1 and adds to -2%2Acos%28theta%29. It may not be obvious, so let's use the quadratic formula.
x=+%28-b%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29
x=%282cos%28theta%29%2B-+sqrt%284cos%5E2%28theta%29+-4%29%29%2F2
x=%282cos%28theta%29%2B-+sqrt%284%28cos%5E2%28theta%29-1%29%29%29%2F2
x=%282cos%28theta%29%2B-+2sqrt%28cos%5E2%28theta%29-1%29%29%2F2
highlight%28x=cos%28theta%29%2B-+sqrt%28cos%5E2%28theta%29-1%29%29
=====================
Let's try completing the square to check:
x%5E2+-2cos%28theta%29x+=+-1
x%5E2+-2cos%28theta%29x+%2B+cos%5E2%28x%29+=+-1+%2B+cos%5E2%28x%29
%28x-cos%28theta%29%29%5E2+=+cos%5E2%28theta%29+-1
Multiply both sides by -1:
-1%2A%28x-cos%28theta%29%5E2%29+=+1-cos%5E2%28theta%29
-1%2A%28x-cos%28theta%29%5E2%29=+sin%5E2%28theta%29
sqrt%28-1%2A%28x-cos%28theta%29%29%5E2%29+=+sqrt%28sin%5E2%28theta%29%29
i%2A%28x-cos%28theta%29%29+=+sin%28theta%29
Multiply both sides by i
-%28x-cos%28theta%29%29+=+0%2B-+i%2Asin%28theta%29
-x+%2Bcos%28theta%29+=+0%2B-+i%2Asin%28theta%29
-x+=+-cos%28theta%29%2B-+i%2Asin%28theta%29
highlight%28x+=+cos%28theta%29%2B-+i%2Asin%28theta%29%29
==========================
ASIDE:
You'll find that if we take the first way a couple steps further we'll get:
x=cos%28theta%29%2B-+sqrt%28-1%2A%281-cos%5E2%28theta%29%29%29
x=cos%28theta%29%2B-+i%2Asqrt%28sin%5E2%28theta%29%29
x=cos%28theta%29%2B-+i%2Asin%28theta%29
Which is what we found by completing the square.
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Thus,
our answer is
highlight%28x=cos%28theta%29%2B-+sqrt%28cos%5E2%28theta%29-1%29%29