Solve the equations:
Q. `x^(5)+242=(243)/(x^(5))`.

Answer

To solve the equation \( x^{5} + 242 = \frac{243}{x^{5}} \), we can follow these steps:

### Step 1: Substitute \( z \) for \( x^{5} \)
Let \( z = x^{5} \). Then, we can rewrite the equation as:
\[ z + 242 = \frac{243}{z} \]

### Step 2: Multiply through by \( z \)
To eliminate the fraction, multiply both sides of the equation by \( z \) (assuming \( z \neq 0 \)):
\[ z^2 + 242z = 243 \]

### Step 3: Rearrange the equation
Rearranging the equation gives us a standard quadratic form:
\[ z^2 + 242z - 243 = 0 \]

### Step 4: Factor the quadratic equation
We need to factor the quadratic equation. We are looking for two numbers that multiply to \(-243\) and add up to \(242\). The numbers are \(243\) and \(-1\). Thus, we can factor the equation as:
\[ (z + 243)(z - 1) = 0 \]

### Step 5: Set each factor to zero
Now, we set each factor equal to zero:
1. \( z + 243 = 0 \)
2. \( z - 1 = 0 \)

### Step 6: Solve for \( z \)
From the first equation:
\[ z = -243 \]
From the second equation:
\[ z = 1 \]

### Step 7: Substitute back to find \( x \)
Recall that \( z = x^{5} \). Now we substitute back to find \( x \).

1. For \( z = -243 \):
\[ x^{5} = -243 \implies x = -3 \quad \text{(since } -243 = -3^{5}\text{)} \]

2. For \( z = 1 \):
\[ x^{5} = 1 \implies x = 1 \quad \text{(since } 1 = 1^{5}\text{)} \]

### Final Solutions
Thus, the solutions for \( x \) are:
\[ x = -3 \quad \text{and} \quad x = 1 \]

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