If a - ` (1)/(a) = 3 , ` find ` a^(2) + 3a +(1)/(a^(2)) - (3)/(a)`

Answer

To solve the problem, we need to find the value of the expression \( a^2 + 3a + \frac{1}{a^2} - 3 \) given that \( a - \frac{1}{a} = 3 \).

### Step-by-step Solution:

1. **Start with the given equation:**
\[ a - \frac{1}{a} = 3 \]

2. **Square both sides of the equation:**
\[ \left(a - \frac{1}{a}\right)^2 = 3^2 \]
This expands to:
\[ a^2 - 2a\left(\frac{1}{a}\right) + \frac{1}{a^2} = 9 \]
Simplifying gives:
\[ a^2 - 2 + \frac{1}{a^2} = 9 \]

3. **Rearranging the equation:**
\[ a^2 + \frac{1}{a^2} = 9 + 2 \]
Therefore:
\[ a^2 + \frac{1}{a^2} = 11 \]

4. **Now substitute into the expression we need to find:**
The expression we need to evaluate is:
\[ a^2 + 3a + \frac{1}{a^2} - 3 \]
We can rewrite this as:
\[ (a^2 + \frac{1}{a^2}) + 3a - 3 \]

5. **Substituting \( a^2 + \frac{1}{a^2} \):**
We already found that \( a^2 + \frac{1}{a^2} = 11 \), so we substitute:
\[ 11 + 3a - 3 \]

6. **Simplifying further:**
\[ 11 - 3 + 3a = 8 + 3a \]

7. **Now, we need to find \( a \):**
We can express \( a \) in terms of \( a - \frac{1}{a} = 3 \):
From \( a - \frac{1}{a} = 3 \), we can rearrange to find \( a \):
\[ a = 3 + \frac{1}{a} \]
Multiplying both sides by \( a \):
\[ a^2 - 3a - 1 = 0 \]
Using the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -3, c = -1 \):
\[ a = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
\[ a = \frac{3 \pm \sqrt{9 + 4}}{2} \]
\[ a = \frac{3 \pm \sqrt{13}}{2} \]

8. **Substituting \( a \) back into \( 8 + 3a \):**
\[ 8 + 3\left(\frac{3 \pm \sqrt{13}}{2}\right) = 8 + \frac{9 \pm 3\sqrt{13}}{2} \]
Converting \( 8 \) to a fraction:
\[ = \frac{16}{2} + \frac{9 \pm 3\sqrt{13}}{2} = \frac{25 \pm 3\sqrt{13}}{2} \]

### Final Answer:
Thus, the value of \( a^2 + 3a + \frac{1}{a^2} - 3 \) is:
\[ \frac{25 \pm 3\sqrt{13}}{2} \]