If `a+bsqrt(5)=(4-3sqrt(5))/(4+3sqrt(5))`, then find the values of a and b.

Explanation

To solve the equation \( a + b\sqrt{5} = \frac{4 - 3\sqrt{5}}{4 + 3\sqrt{5}} \), we will first rationalize the right-hand side (RHS).

### Step 1: Rationalizing the RHS
We need to multiply the numerator and denominator of the RHS by the conjugate of the denominator, which is \( 4 - 3\sqrt{5} \).

\[ \text{RHS} = \frac{(4 - 3\sqrt{5})(4 - 3\sqrt{5})}{(4 + 3\sqrt{5})(4 - 3\sqrt{5})} \]

### Step 2: Expanding the Numerator
Now, we will expand the numerator:

\[ (4 - 3\sqrt{5})^2 = 4^2 - 2 \cdot 4 \cdot 3\sqrt{5} + (3\sqrt{5})^2 = 16 - 24\sqrt{5} + 45 = 61 - 24\sqrt{5} \]

### Step 3: Expanding the Denominator
Next, we will expand the denominator using the difference of squares:

\[ (4 + 3\sqrt{5})(4 - 3\sqrt{5}) = 4^2 - (3\sqrt{5})^2 = 16 - 45 = -29 \]

### Step 4: Putting it Together
Now we can write the RHS as:

\[ \text{RHS} = \frac{61 - 24\sqrt{5}}{-29} = -\frac{61}{29} + \frac{24\sqrt{5}}{29} \]

### Step 5: Comparing LHS and RHS
Now we have:

\[ a + b\sqrt{5} = -\frac{61}{29} + \frac{24}{29}\sqrt{5} \]

From this, we can compare coefficients:

- For the constant term (without \(\sqrt{5}\)), we have:
\[ a = -\frac{61}{29} \]

- For the coefficient of \(\sqrt{5}\), we have:
\[ b = \frac{24}{29} \]

### Final Values
Thus, the values of \( a \) and \( b \) are:

\[ a = -\frac{61}{29}, \quad b = \frac{24}{29} \]

---