The equation `x^((3)/4 (log_2x)^2+log_2 x -(5)/(4))=sqrt(2)` has :

Options

A. atleast one-real solution
B. exactly three - real solutions
C. exactly one irrational solution
D. complex roots

Answer

To solve the equation

\[ x^{\left(\frac{3}{4}(\log_2 x)^2 + \log_2 x - \frac{5}{4}\right)} = \sqrt{2}, \]

we will follow these steps:

### Step 1: Take logarithm of both sides
We take the logarithm (base 2) of both sides:

\[ \log_2\left(x^{\left(\frac{3}{4}(\log_2 x)^2 + \log_2 x - \frac{5}{4}\right)}\right) = \log_2(\sqrt{2}). \]

### Step 2: Simplify the logarithmic expression
Using the property of logarithms that states \(\log_b(a^c) = c \cdot \log_b(a)\), we can simplify the left side:

\[ \left(\frac{3}{4}(\log_2 x)^2 + \log_2 x - \frac{5}{4}\right) \cdot \log_2 x = \frac{1}{2}. \]

### Step 3: Set \(t = \log_2 x\)
Let \(t = \log_2 x\). Then the equation becomes:

\[ \left(\frac{3}{4}t^2 + t - \frac{5}{4}\right) t = \frac{1}{2}. \]

### Step 4: Rearrange the equation
Multiply both sides by 4 to eliminate the fraction:

\[ 3t^3 + 4t^2 - 5t - 2 = 0. \]

### Step 5: Use the Rational Root Theorem
We will use trial and error (Rational Root Theorem) to find rational roots. Let's test \(t = 1\):

\[ 3(1)^3 + 4(1)^2 - 5(1) - 2 = 3 + 4 - 5 - 2 = 0. \]

Thus, \(t = 1\) is a root.

### Step 6: Factor the polynomial
Now we can factor the polynomial \(3t^3 + 4t^2 - 5t - 2\) by dividing it by \(t - 1\):

Using synthetic division, we find:

\[ 3t^3 + 4t^2 - 5t - 2 = (t - 1)(3t^2 + 7t + 2). \]

### Step 7: Solve the quadratic equation
Now, we solve the quadratic equation \(3t^2 + 7t + 2 = 0\) using the quadratic formula:

\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3} = \frac{-7 \pm \sqrt{49 - 24}}{6} = \frac{-7 \pm \sqrt{25}}{6} = \frac{-7 \pm 5}{6}. \]

This gives us:

\[ t = \frac{-2}{6} = -\frac{1}{3} \quad \text{and} \quad t = \frac{-12}{6} = -2. \]

### Step 8: Find the values of \(x\)
Now we have three values for \(t\):
1. \(t = 1\) implies \(\log_2 x = 1 \Rightarrow x = 2\).
2. \(t = -\frac{1}{3}\) implies \(\log_2 x = -\frac{1}{3} \Rightarrow x = 2^{-\frac{1}{3}} = \frac{1}{\sqrt[3]{2}}\).
3. \(t = -2\) implies \(\log_2 x = -2 \Rightarrow x = 2^{-2} = \frac{1}{4}\).

### Conclusion
The solutions for \(x\) are:
- \(x = 2\) (rational)
- \(x = \frac{1}{4}\) (rational)
- \(x = \frac{1}{\sqrt[3]{2}}\) (irrational)

### Summary of Solutions
1. At least one real solution: **True** (we have three).
2. Exactly three real solutions: **True** (we found three).
3. Exactly one irrational solution: **True** (only one irrational solution).
4. Complex roots: **False** (all roots are real).