Certainly! Let's solve the given expression step-by-step:
The expression to be evaluated is:
[tex]\[
\frac{18 + \sqrt{18^2 + 4 \cdot 6 \cdot 18}}{2 \cdot 6}
\][/tex]
### Step 1: Calculate the Discriminant
First, let's compute the value inside the square root:
[tex]\[
18^2 = 324
\][/tex]
[tex]\[
4 \cdot 6 \cdot 18 = 432
\][/tex]
Adding these together:
[tex]\[
18^2 + 4 \cdot 6 \cdot 18 = 324 + 432 = 756
\][/tex]
### Step 2: Calculate the Square Root
Next, compute the square root of 756:
[tex]\[
\sqrt{756} \approx 27.49545416973504
\][/tex]
### Step 3: Calculate the Numerator
Add this square root value to 18:
[tex]\[
18 + 27.49545416973504 = 45.49545416973504
\][/tex]
### Step 4: Calculate the Denominator
Calculate the denominator:
[tex]\[
2 \cdot 6 = 12
\][/tex]
### Step 5: Divide the Numerator by the Denominator
Finally, divide the result of the numerator by the denominator:
[tex]\[
\frac{45.49545416973504}{12} \approx 3.79128784747792
\][/tex]
### Conclusion
Hence, the value of the given expression is:
[tex]\[
\frac{18 + \sqrt{18^2 + 4 \cdot 6 \cdot 18}}{2 \cdot 6} \approx 3.79128784747792
\][/tex]
