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

Options

A. at least 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 can follow these steps:

### Step 1: Rewrite the equation using logarithmic properties
We start by rewriting \( \sqrt{2} \) as \( 2^{1/2} \):
\[ x^{\left(\frac{3}{4}(\log_2 x)^2 + \log_2 x - \frac{5}{4}\right)} = 2^{1/2} \]

### Step 2: Take logarithm base 2 of both sides
Taking the logarithm base 2 of both sides gives us:
\[ \left(\frac{3}{4}(\log_2 x)^2 + \log_2 x - \frac{5}{4}\right) \log_2 x = \frac{1}{2} \]

### Step 3: Let \( y = \log_2 x \)
Substituting \( y = \log_2 x \), we rewrite the equation:
\[ \left(\frac{3}{4}y^2 + y - \frac{5}{4}\right) y = \frac{1}{2} \]

### Step 4: Simplify the equation
Expanding the left side:
\[ \frac{3}{4}y^3 + y^2 - \frac{5}{4}y = \frac{1}{2} \]
Multiplying through by 4 to eliminate the fraction:
\[ 3y^3 + 4y^2 - 5y = 2 \]
Rearranging gives:
\[ 3y^3 + 4y^2 - 5y - 2 = 0 \]

### Step 5: Factor the polynomial
We can factor the polynomial:
\[ 3y^3 + 4y^2 - 5y - 2 = 0 \]
We can use the Rational Root Theorem or synthetic division to find possible rational roots. Testing \( y = 1 \):
\[ 3(1)^3 + 4(1)^2 - 5(1) - 2 = 3 + 4 - 5 - 2 = 0 \]
Thus, \( y - 1 \) is a factor. We can perform polynomial long division:
\[ 3y^3 + 4y^2 - 5y - 2 = (y - 1)(3y^2 + 7y + 2) \]

### Step 6: Solve the quadratic equation
Now we solve \( 3y^2 + 7y + 2 = 0 \) using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3} = \frac{-7 \pm \sqrt{49 - 24}}{6} = \frac{-7 \pm \sqrt{25}}{6} \]
\[ y = \frac{-7 \pm 5}{6} \]
This gives us two solutions:
\[ y_1 = \frac{-2}{6} = -\frac{1}{3}, \quad y_2 = \frac{-12}{6} = -2 \]

### Step 7: Find values of \( x \)
Now we revert back to \( x \):
1. For \( y = 1 \):
\[ \log_2 x = 1 \implies x = 2 \]
2. For \( y = -\frac{1}{3} \):
\[ \log_2 x = -\frac{1}{3} \implies x = 2^{-\frac{1}{3}} = \frac{1}{\sqrt[3]{2}} \]
3. For \( y = -2 \):
\[ \log_2 x = -2 \implies x = 2^{-2} = \frac{1}{4} \]

### Final Solutions
The solutions to the original equation are:
\[ x = 2, \quad x = \frac{1}{\sqrt[3]{2}}, \quad x = \frac{1}{4} \]