The equation `(sqrt(5)+2)^(|x|)+(sqrt(5)-2)^(|x|)=18`. has

Options

A. only one solution
B. two solutions
C. no solution
D. any number of solutions

Explanation

To solve the equation \((\sqrt{5}+2)^{|x|}+(\sqrt{5}-2)^{|x|}=18\), we will follow these steps:

### Step 1: Substitute \(y = |x|\)
Let \(y = |x|\). The equation then becomes:
\[ (\sqrt{5}+2)^{y}+(\sqrt{5}-2)^{y}=18 \]

### Step 2: Analyze the terms
We note that \((\sqrt{5}+2) > 1\) and \((\sqrt{5}-2) < 1\). This means that as \(y\) increases, \((\sqrt{5}+2)^{y}\) will increase while \((\sqrt{5}-2)^{y}\) will decrease.

### Step 3: Check for specific values of \(y\)
We can check specific values of \(y\) to see if they satisfy the equation.

1. **For \(y = 2\)**:
\[ (\sqrt{5}+2)^{2} + (\sqrt{5}-2)^{2} \]
Calculate \((\sqrt{5}+2)^{2}\):
\[ = (\sqrt{5})^{2} + 2 \cdot \sqrt{5} \cdot 2 + 2^{2} = 5 + 4\sqrt{5} + 4 = 9 + 4\sqrt{5} \]
Calculate \((\sqrt{5}-2)^{2}\):
\[ = (\sqrt{5})^{2} - 2 \cdot \sqrt{5} \cdot 2 + 2^{2} = 5 - 4\sqrt{5} + 4 = 9 - 4\sqrt{5} \]
Now add these two results:
\[ (9 + 4\sqrt{5}) + (9 - 4\sqrt{5}) = 18 \]
This shows that \(y = 2\) is indeed a solution.

### Step 4: Determine possible values of \(x\)
Since \(y = |x|\), we have:
\[ |x| = 2 \implies x = 2 \text{ or } x = -2 \]

### Step 5: Count the solutions
Thus, there are two solutions for \(x\):
\[ x = 2 \quad \text{and} \quad x = -2 \]

### Conclusion
The equation \((\sqrt{5}+2)^{|x|}+(\sqrt{5}-2)^{|x|}=18\) has a total of **2 solutions**.

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