solve 1-(2/x)=2/(x+2)

Express your answer in simplest radical form.

What is radical form ne how?

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

Multiply both sides by x(x+2):

x(x+2) - 2(x+2) = 2x

Note that we cannot allow x to be 0 or minus two because for these values the original equation is not defined. The left hand side is:

x(x+2) - 2(x+2).

Factor out the term (x + 2):

x(x+2) - 2(x+2) = (x-2)(x+2) = x^2 - 4

If you now bring the term 2x from the right hand side to the left you get the equation:

x^2 - 2x - 4 = 0 --->

x = 1 +/- sqrt[5]

To solve the equation 1 - (2/x) = 2/(x+2) and express the answer in simplest radical form, follow these steps:

Step 1: Multiply both sides of the equation by x(x+2) to eliminate the denominators:

x(x+2) - 2(x+2) = 2x

This step is necessary to simplify the equation and remove the fractions.

Step 2: Simplify the left-hand side of the equation:

x(x+2) - 2(x+2) = (x-2)(x+2) = x^2 - 4

By distributing, the expression simplifies to x^2 - 4.

Step 3: Move the 2x term from the right-hand side to the left-hand side:

x^2 - 2x - 4 = 0

Step 4: Solve the quadratic equation above. In this case, you can try factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:

x = (-b +/- sqrt(b^2 - 4ac)) / (2a)

In the equation x^2 - 2x - 4 = 0, a = 1, b = -2, and c = -4.

x = (-(-2) +/- sqrt((-2)^2 - 4(1)(-4))) / (2(1))

x = (2 +/- sqrt(4 + 16)) / 2

x = (2 +/- sqrt(20)) / 2

Step 5: Simplify the expression inside the square root:

sqrt(20) = sqrt(4 * 5) = 2sqrt(5)

Step 6: Substitute the simplified expression for the square root back into the equation:

x = (2 +/- 2sqrt(5)) / 2

Step 7: Simplify the expression by canceling out the common factor of 2 in the numerator and denominator:

x = 1 +/- sqrt(5)

Therefore, the solutions to the equation 1 - (2/x) = 2/(x+2) in simplest radical form are x = 1 + sqrt(5) and x = 1 - sqrt(5).