Question
Solve the rational equation
x^2 +x/x^2-5x+4 = 4/x-4 +4/x^2 -5x+4
x^2 +x/x^2-5x+4 = 4/x-4 +4/x^2 -5x+4
Answers
GPT-4o mini
To solve the rational equation
\[
\frac{x^2 + x}{x^2 - 5x + 4} = \frac{4}{x - 4} + \frac{4}{x^2 - 5x + 4},
\]
we will first simplify and find a common denominator for the right side.
### Step 1: Simplify the right side
The denominator of the left-hand side is \(x^2 - 5x + 4\). We can factor this expression:
\[
x^2 - 5x + 4 = (x-1)(x-4).
\]
Next, rewrite the fractions on the right-hand side:
\[
\frac{4}{x - 4} + \frac{4}{(x - 1)(x - 4)}.
\]
To combine these fractions, we need a common denominator, which is \((x - 1)(x - 4)\). Therefore, the first term can be rewritten as follows:
\[
\frac{4}{x - 4} = \frac{4(x - 1)}{(x - 1)(x - 4)} = \frac{4x - 4}{(x - 1)(x - 4)}.
\]
Now we can add the two fractions:
\[
\frac{4x - 4 + 4}{(x - 1)(x - 4)} = \frac{4x}{(x - 1)(x - 4)}.
\]
### Step 2: Write the equality
Now, our equation becomes:
\[
\frac{x^2 + x}{(x - 1)(x - 4)} = \frac{4x}{(x - 1)(x - 4)}.
\]
### Step 3: Eliminate the denominators
Since the denominators are the same and we assume they are not zero (we will check restrictions on \(x\) later), we can equate the numerators:
\[
x^2 + x = 4x.
\]
### Step 4: Rearrange the equation
Rearranging gives us:
\[
x^2 + x - 4x = 0 \implies x^2 - 3x = 0.
\]
Factoring the left-hand side gives:
\[
x(x - 3) = 0.
\]
### Step 5: Find the solutions
Setting each factor to zero gives:
\[
x = 0 \quad \text{or} \quad x = 3.
\]
### Step 6: Check for restrictions
Now we must check for any restrictions on \(x\) based on the original denominators:
1. **From** \(x^2 - 5x + 4\):
Set \(x^2 - 5x + 4 = 0\):
\[
(x - 1)(x - 4) = 0 \implies x = 1 \text{ or } x = 4.
\]
Therefore, \(x \neq 1\) and \(x \neq 4\).
2. **From** \(x - 4\):
This already puts a restriction that \(x \neq 4\).
### Conclusion
The possible solutions \(x = 0\) and \(x = 3\) do not violate these restrictions.
Thus, the solutions to the rational equation are:
\[
\boxed{0 \text{ and } 3}.
\]
\[
\frac{x^2 + x}{x^2 - 5x + 4} = \frac{4}{x - 4} + \frac{4}{x^2 - 5x + 4},
\]
we will first simplify and find a common denominator for the right side.
### Step 1: Simplify the right side
The denominator of the left-hand side is \(x^2 - 5x + 4\). We can factor this expression:
\[
x^2 - 5x + 4 = (x-1)(x-4).
\]
Next, rewrite the fractions on the right-hand side:
\[
\frac{4}{x - 4} + \frac{4}{(x - 1)(x - 4)}.
\]
To combine these fractions, we need a common denominator, which is \((x - 1)(x - 4)\). Therefore, the first term can be rewritten as follows:
\[
\frac{4}{x - 4} = \frac{4(x - 1)}{(x - 1)(x - 4)} = \frac{4x - 4}{(x - 1)(x - 4)}.
\]
Now we can add the two fractions:
\[
\frac{4x - 4 + 4}{(x - 1)(x - 4)} = \frac{4x}{(x - 1)(x - 4)}.
\]
### Step 2: Write the equality
Now, our equation becomes:
\[
\frac{x^2 + x}{(x - 1)(x - 4)} = \frac{4x}{(x - 1)(x - 4)}.
\]
### Step 3: Eliminate the denominators
Since the denominators are the same and we assume they are not zero (we will check restrictions on \(x\) later), we can equate the numerators:
\[
x^2 + x = 4x.
\]
### Step 4: Rearrange the equation
Rearranging gives us:
\[
x^2 + x - 4x = 0 \implies x^2 - 3x = 0.
\]
Factoring the left-hand side gives:
\[
x(x - 3) = 0.
\]
### Step 5: Find the solutions
Setting each factor to zero gives:
\[
x = 0 \quad \text{or} \quad x = 3.
\]
### Step 6: Check for restrictions
Now we must check for any restrictions on \(x\) based on the original denominators:
1. **From** \(x^2 - 5x + 4\):
Set \(x^2 - 5x + 4 = 0\):
\[
(x - 1)(x - 4) = 0 \implies x = 1 \text{ or } x = 4.
\]
Therefore, \(x \neq 1\) and \(x \neq 4\).
2. **From** \(x - 4\):
This already puts a restriction that \(x \neq 4\).
### Conclusion
The possible solutions \(x = 0\) and \(x = 3\) do not violate these restrictions.
Thus, the solutions to the rational equation are:
\[
\boxed{0 \text{ and } 3}.
\]