If `2a-(2)/(a)+3=0`, then value of `(a^(3)-(1)/(a^(3))+2)` is -

Options

A. 5
B. `-(35)/(8)`
C. `-(40)/(7)`
D. `(-47)/(8)`

Answer

To solve the equation \(2a - \frac{2}{a} + 3 = 0\) and find the value of \(a^3 - \frac{1}{a^3} + 2\), we can follow these steps:

### Step 1: Rearranging the Equation
Start with the given equation:
\[ 2a - \frac{2}{a} + 3 = 0 \]
Rearranging gives:
\[ 2a + 3 = \frac{2}{a} \]

### Step 2: Eliminate the Fraction
Multiply both sides by \(a\) to eliminate the fraction:
\[ a(2a + 3) = 2 \]
This simplifies to:
\[ 2a^2 + 3a - 2 = 0 \]

### Step 3: Factor the Quadratic Equation
Now we need to factor the quadratic equation \(2a^2 + 3a - 2 = 0\). We can look for two numbers that multiply to \(2 \times -2 = -4\) and add to \(3\). The numbers \(4\) and \(-1\) work:
\[ 2a^2 + 4a - a - 2 = 0 \]
Grouping gives:
\[ (2a^2 + 4a) + (-a - 2) = 0 \]
Factoring out common terms:
\[ 2a(a + 2) - 1(a + 2) = 0 \]
This can be factored as:
\[ (2a - 1)(a + 2) = 0 \]

### Step 4: Solve for \(a\)
Setting each factor to zero gives us:
1. \(2a - 1 = 0 \implies a = \frac{1}{2}\)
2. \(a + 2 = 0 \implies a = -2\)

### Step 5: Calculate \(a^3 - \frac{1}{a^3} + 2\)
Now we will calculate \(a^3 - \frac{1}{a^3} + 2\) for both values of \(a\).

#### Case 1: \(a = -2\)
Calculate \(a^3\):
\[ a^3 = (-2)^3 = -8 \]
Calculate \(\frac{1}{a^3}\):
\[ \frac{1}{a^3} = \frac{1}{-8} = -\frac{1}{8} \]
Now substitute into the expression:
\[ -8 - \left(-\frac{1}{8}\right) + 2 = -8 + \frac{1}{8} + 2 \]
Convert \(-8\) to a fraction:
\[ -8 = -\frac{64}{8} \]
So:
\[ -\frac{64}{8} + \frac{1}{8} + \frac{16}{8} = -\frac{64 + 1 - 16}{8} = -\frac{47}{8} \]

#### Case 2: \(a = \frac{1}{2}\)
Calculate \(a^3\):
\[ a^3 = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]
Calculate \(\frac{1}{a^3}\):
\[ \frac{1}{a^3} = 8 \]
Now substitute into the expression:
\[ \frac{1}{8} - 8 + 2 = \frac{1}{8} - \frac{64}{8} + \frac{16}{8} = \frac{1 - 64 + 16}{8} = \frac{-47}{8} \]

### Final Answer
In both cases, we find:
\[ a^3 - \frac{1}{a^3} + 2 = -\frac{47}{8} \]