If `a ne b and a^(2)=5a-3,b^(2)=5b-3`, then form that equation whose roots are `a/b and b/a`.

Answer

To solve the problem, we need to find the quadratic equation whose roots are \( \frac{a}{b} \) and \( \frac{b}{a} \) given the equations \( a^2 = 5a - 3 \) and \( b^2 = 5b - 3 \).

### Step-by-Step Solution:

1. **Write down the equations**:
We have two equations:
\[ a^2 = 5a - 3 \quad \text{(1)} \]
\[ b^2 = 5b - 3 \quad \text{(2)} \]

2. **Subtract the two equations**:
Subtract equation (2) from equation (1):
\[ a^2 - b^2 = (5a - 3) - (5b - 3) \]
Simplifying this gives:
\[ a^2 - b^2 = 5a - 5b \]
We can factor the left-hand side:
\[ (a - b)(a + b) = 5(a - b) \]
Since \( a \neq b \), we can divide both sides by \( a - b \):
\[ a + b = 5 \quad \text{(3)} \]

3. **Add the two equations**:
Now, add equations (1) and (2):
\[ a^2 + b^2 = (5a - 3) + (5b - 3) \]
This simplifies to:
\[ a^2 + b^2 = 5a + 5b - 6 \]
Using equation (3), substitute \( a + b = 5 \):
\[ a^2 + b^2 = 5(5) - 6 = 25 - 6 = 19 \quad \text{(4)} \]

4. **Calculate \( ab \)**:
We can use the identity \( (a + b)^2 = a^2 + b^2 + 2ab \):
\[ 5^2 = 19 + 2ab \]
This gives:
\[ 25 = 19 + 2ab \]
Rearranging gives:
\[ 2ab = 25 - 19 = 6 \implies ab = 3 \quad \text{(5)} \]

5. **Form the quadratic equation**:
The roots of the quadratic equation we want are \( \frac{a}{b} \) and \( \frac{b}{a} \). The sum of the roots is:
\[ \frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab} = \frac{19}{3} \]
The product of the roots is:
\[ \frac{a}{b} \cdot \frac{b}{a} = 1 \]
Therefore, the quadratic equation is:
\[ x^2 - \left(\frac{19}{3}\right)x + 1 = 0 \]

6. **Clear the fraction**:
To eliminate the fraction, multiply through by 3:
\[ 3x^2 - 19x + 3 = 0 \]

### Final Answer:
The required quadratic equation is:
\[ 3x^2 - 19x + 3 = 0 \]