Solve (1/x^2)-(4/x)+1=0

Question

Solve the equation.

Answer 1

I don't know that this is the right way to answer this question, but here is what I found.

 

You start with:

 

(1/x²) - (4/x) + 1 = 0

 

What I did was factor out, from all of the terms, 1/x² like this:

 

1/x²(1 - 4x + x²) = 0

 

Now, we know that this equation must equal 0, so one of those terms has to equal 0, and there is nothing that you can plug in for x to make 1/x² equal to 0, so that must mean (1 - 4x + x²) must equal 0.

 

You can change the terms around so it becomes:

 

x² - 4x +1 = 0

 

And plug into the quadratic formula:

 

x =  -b ± √(b² - 4ac) 

               2a

 

x =  -(-4) ± √(16 - 4(1)(1)

                   2(1)

 

x = 3.732, 0.268

Answer 2

One way to do this problem is to manipulate it so you have it in the form 0=ax^2+bx+c

1/x^2-4/x+1=0

1/x^2=4/x+1

 Multiply the x^2 across

1=4x+x^2

x^2+4x-1=0

 

Now use the quadratic formula

-b±sqrt(b^2-4ac)/2a

 

-(4)+sqrt(4^2-4(1)(-1))/[2(2)]

x=2+sqrt(3)

 

-(4)-sqrt(4^2-4(1)(-1))/[2(2)]

x=2-sqrt(3)