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).