`x ^(2) - (6sqrt5)x=-40`
What is the sum of the possible values of x given the above equation ?

Options

A. 15
B. `5 sqrt5`
C. `6 sqrt5`
D. 60

Explanation

To solve the equation \( x^2 - 6\sqrt{5}x = -40 \) and find the sum of the possible values of \( x \), we can follow these steps:

### Step 1: Rearrange the Equation
First, we will move \(-40\) to the left side of the equation to set it to zero:
\[ x^2 - 6\sqrt{5}x + 40 = 0 \]

### Step 2: Identify Coefficients
Now, we identify the coefficients \( a \), \( b \), and \( c \) from the standard form of a quadratic equation \( ax^2 + bx + c = 0 \):
- \( a = 1 \)
- \( b = -6\sqrt{5} \)
- \( c = 40 \)

### Step 3: Use the Quadratic Formula
The quadratic formula to find the roots of the equation is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ x = \frac{-(-6\sqrt{5}) \pm \sqrt{(-6\sqrt{5})^2 - 4 \cdot 1 \cdot 40}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{6\sqrt{5} \pm \sqrt{36 \cdot 5 - 160}}{2} \]

### Step 4: Simplify the Expression Under the Square Root
Calculating \( 36 \cdot 5 - 160 \):
\[ 36 \cdot 5 = 180 \]
So,
\[ 180 - 160 = 20 \]
Thus, we have:
\[ x = \frac{6\sqrt{5} \pm \sqrt{20}}{2} \]

### Step 5: Simplify \( \sqrt{20} \)
We can simplify \( \sqrt{20} \):
\[ \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} \]
Substituting this back into our equation:
\[ x = \frac{6\sqrt{5} \pm 2\sqrt{5}}{2} \]

### Step 6: Split into Two Possible Solutions
This gives us two possible values for \( x \):
\[ x = \frac{(6\sqrt{5} + 2\sqrt{5})}{2} = \frac{8\sqrt{5}}{2} = 4\sqrt{5} \]
and
\[ x = \frac{(6\sqrt{5} - 2\sqrt{5})}{2} = \frac{4\sqrt{5}}{2} = 2\sqrt{5} \]

### Step 7: Find the Sum of the Possible Values
Now, we need to find the sum of the two possible values of \( x \):
\[ 4\sqrt{5} + 2\sqrt{5} = 6\sqrt{5} \]

### Final Answer
Thus, the sum of the possible values of \( x \) is:
\[ \boxed{6\sqrt{5}} \]