Help with this problem?

Question

What are the solutions to the equation:

1 + 1/x² = 2/x

Confusing... incase it is it reads: One plus one over x squared equals two over x.
How to start?

Answer 1

1 + 1/x² = 2/x

x²(1) + x²(1/x²) = x²(2/x)

x² + 1 = 2x

x² - 2x + 1 = 2x - 2x

x² - 2x + 1 = 0

The middle term is - 2x

Find the sum of the middle term

x = 1

Multiply the first term 1 times the last term 1 equals 1 and factor

The factors of 1 =

1 x 1. . .<=. . use these factors

- 1 aand - 1 satisfy the sum of the middle term

insert - x and - x into the equation

x² - 2x + 1 = 0

Factor by grouping

x² - x - x + 1

x(x - 1) - 1(x - 1)

(x - 1)(x - 1)

- - - - - - - s-

Answer 2

If it is (1+1)/x^2=2/x The answer is 1. Because (1+1)/x^2 is simplified to 2/x^2 so you can take the 2/ part out of each side because they are equal and are left with x^2=x and the only solution to that is 1.

Answer 3

Multiply both sides by x^2 gives you:

x^2 + 1 = (2x^2)/x

Simplifying the RHS gets you:

x^2+1=2x

Subtracting 2x from both sides gives you

x^2-2x+1=0

factorising gives you

(x-1)^2=0

Which is true if and only if (x-1)=0 which implies x=1.

This is an example of a repeated root, so technically speaking you have two roots, both of which are x=1.

Answer 4

This problem will have solutions as long as x ≠ 0, because you can't have zero as a denominator of a fraction.

Multiply through by a common denominator:
x²(1 + 1/x²) = x²(2/x)
x² + 1 = 2x
x² - 2x + 1 = 0
(x - 1)² = 0
x - 1 = 0
x = 1.

Answer 5

it looks confusing but it really isn't. first, multiply both sides by (x^2). if you do that step, you no longer have any denominators. after that you should have the equation (x^2 + 1 = 2x). from that point on, its just a normal quadratic equation problem. put the equation in the form (x^2 - 2x +1 =0) and use the quadratice formula.

Answer 6

=7/3

♥

Answer 7

Multiply right through by x squared, then it becomes -
x squared +1 = 2x then
x squared -2x+1= zero then
(x-1) x (x-1) = zero so
x=1

Answer 8

multiply both sides by x^(power)2
then you have x^2+1=2x
just bring the 2x over
the rest is easy

Answer 9

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

Answer 10

1 + 1/x2 = 2/x

x2/x2 + 1/x2 = 2/x

(x2 + 1)/ x2 = 2/x

x (x2 + 1) = 2 x2 (cross multiply)

x3 + x = 2 x2

x (x2 + 1 – 2x) = 0

x (x + 2) (x-1) =0

x = 0, -2, 1