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Solve the rational equation:

2
z²+1 + x = 2 + z²1·
x-1
A. x = 1
B. x =
C. x = 1, x = 1
D. There is no solution.

User Davosmith
by
8.7k points

1 Answer

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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

User MichelZ
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