If `(15+4sqrt(14))^(t)+(15-4sqrt(14))^(t)=30` where `t=x^(2)-2|x|` , then the value of x= ….......

Answer

To solve the equation \( (15 + 4\sqrt{14})^t + (15 - 4\sqrt{14})^t = 30 \) where \( t = x^2 - 2|x| \), we can follow these steps:

### Step 1: Define Variables
Let:
\[ a = 15 + 4\sqrt{14} \]
\[ b = 15 - 4\sqrt{14} \]

Thus, we can rewrite the equation as:
\[ a^t + b^t = 30 \]

### Step 2: Analyze the Terms
Notice that:
\[ b = \frac{1}{a} \]
because \( a \cdot b = (15 + 4\sqrt{14})(15 - 4\sqrt{14}) = 15^2 - (4\sqrt{14})^2 = 225 - 224 = 1 \).

This means:
\[ b^t = \left(\frac{1}{a}\right)^t = \frac{1}{a^t} \]

### Step 3: Substitute in the Equation
Substituting \( b^t \) into the equation gives:
\[ a^t + \frac{1}{a^t} = 30 \]

### Step 4: Let \( p = a^t \)
Let \( p = a^t \). Then the equation becomes:
\[ p + \frac{1}{p} = 30 \]

### Step 5: Multiply through by \( p \)
Multiply both sides by \( p \):
\[ p^2 + 1 = 30p \]
Rearranging gives:
\[ p^2 - 30p + 1 = 0 \]

### Step 6: Solve the Quadratic Equation
Using the quadratic formula \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -30, c = 1 \):
\[ p = \frac{30 \pm \sqrt{(-30)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ p = \frac{30 \pm \sqrt{900 - 4}}{2} \]
\[ p = \frac{30 \pm \sqrt{896}}{2} \]
\[ p = \frac{30 \pm 4\sqrt{14}}{2} \]
\[ p = 15 \pm 2\sqrt{14} \]

### Step 7: Find Values of \( t \)
We have two cases for \( p \):
1. \( a^t = 15 + 2\sqrt{14} \)
2. \( a^t = 15 - 2\sqrt{14} \)

#### Case 1: \( a^t = 15 + 2\sqrt{14} \)
Taking logarithm:
\[ t \log(15 + 4\sqrt{14}) = \log(15 + 2\sqrt{14}) \]
Thus:
\[ t = \frac{\log(15 + 2\sqrt{14})}{\log(15 + 4\sqrt{14})} \]

#### Case 2: \( a^t = 15 - 2\sqrt{14} \)
Similarly:
\[ t \log(15 + 4\sqrt{14}) = \log(15 - 2\sqrt{14}) \]
Thus:
\[ t = \frac{\log(15 - 2\sqrt{14})}{\log(15 + 4\sqrt{14})} \]

### Step 8: Relate \( t \) to \( x \)
Now, we know \( t = x^2 - 2|x| \). We can set:
1. \( x^2 - 2|x| = 1 \)
2. \( x^2 - 2|x| = -1 \)

### Step 9: Solve for \( x \)
1. **For \( x^2 - 2|x| = 1 \)**:
- If \( x \geq 0 \): \( x^2 - 2x - 1 = 0 \) → \( x = 1 \pm \sqrt{2} \)
- If \( x < 0 \): \( x^2 + 2x - 1 = 0 \) → \( x = -1 \pm \sqrt{2} \)

2. **For \( x^2 - 2|x| = -1 \)**:
- If \( x \geq 0 \): \( x^2 - 2x + 1 = 0 \) → \( x = 1 \)
- If \( x < 0 \): \( x^2 + 2x + 1 = 0 \) → \( x = -1 \)

### Final Values of \( x \)
Thus, the possible values of \( x \) are:
\[ x = 1, -1, 1 + \sqrt{2}, 1 - \sqrt{2}, -1 + \sqrt{2}, -1 - \sqrt{2} \]

### Summary of Solutions
The values of \( x \) are:
\[ x = \pm 1, \pm (1 + \sqrt{2}), \pm (1 - \sqrt{2}) \]