Suppose `a, b in Q and 3 - sqrt(5)` is a root of `x^(2) + ax + b = 0 , "then" a^(2) - 4b - 18 . 8` = ________

Answer

To solve the problem, we need to find the value of \( a^2 - 4b - 18.8 \) given that \( 3 - \sqrt{5} \) is a root of the quadratic equation \( x^2 + ax + b = 0 \), where \( a \) and \( b \) are rational numbers.

### Step 1: Identify the roots
Since \( 3 - \sqrt{5} \) is a root and \( a \) and \( b \) are rational, the other root must be the conjugate, \( 3 + \sqrt{5} \). Thus, the roots are:
- \( r_1 = 3 - \sqrt{5} \)
- \( r_2 = 3 + \sqrt{5} \)

### Step 2: Calculate the sum of the roots
The sum of the roots \( r_1 + r_2 \) is:
\[ (3 - \sqrt{5}) + (3 + \sqrt{5}) = 6 \]
According to Vieta's formulas, the sum of the roots is also given by \( -\frac{a}{1} \), so:
\[ -\frac{a}{1} = 6 \implies a = -6 \]

### Step 3: Calculate the product of the roots
The product of the roots \( r_1 \cdot r_2 \) is:
\[ (3 - \sqrt{5})(3 + \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4 \]
According to Vieta's formulas, the product of the roots is also given by \( \frac{b}{1} \), so:
\[ \frac{b}{1} = 4 \implies b = 4 \]

### Step 4: Substitute \( a \) and \( b \) into the expression
Now that we have \( a = -6 \) and \( b = 4 \), we can substitute these values into the expression \( a^2 - 4b - 18.8 \):
\[ a^2 - 4b - 18.8 = (-6)^2 - 4(4) - 18.8 \]
Calculating each term:
\[ = 36 - 16 - 18.8 \]
\[ = 36 - 34.8 \]
\[ = 1.2 \]

### Final Answer
Thus, the value of \( a^2 - 4b - 18.8 \) is:
\[ \boxed{1.2} \]