Final answer:
To solve the rational equation, we simplify the equation, combine like terms, and solve a quadratic equation in terms of x.
Step-by-step explanation:
Let's solve the rational equation:
(2/(z^2+1)) + x = 2 + (z^2)/(x-1)
First, we want to simplify the equation by finding a common denominator. The common denominator here is (z^2+1)(x-1). So, let's multiply every term in the equation by (z^2+1)(x-1).
(2(x-1))/(z^2+1) + x(x-1) = 2(z^2+1) + z^2
Next, we can simplify the equation further by distributing and combining like terms.
2x - 2 + x^2 - x + 2z^2 + 2 = 2z^2 + 2 + z^2
Combining like terms again, we have:
x^2 + x - 2 + 2z^2 - z^2 - 2 = 0
Simplifying further:
x^2 + x - z^2 - 4 = 0
Now, we have a quadratic equation in terms of x. We can solve this equation by factoring or using the quadratic formula.
Factoring this equation, we get:
(x - 1)(x + 2) - (z - 2)(z + 2) = 0
From here, we can set each factor equal to zero and solve for x:
x - 1 = 0 --> x = 1
x + 2 = 0 --> x = -2
So, the solutions to the rational equation are x = 1 and x = -2.
Learn more about Solving rational equations