What is the sum of the solutions of
`(6x +5) ^(2) - (3x -2)^(2) =0` ?

Options

A. `-8/3`
B. `-1/6`
C. `7/3`
D. 3

Explanation

To solve the equation \((6x + 5)^2 - (3x - 2)^2 = 0\) and find the sum of the solutions, we can follow these steps:

### Step 1: Set the equation to zero
We start with the equation:
\[ (6x + 5)^2 - (3x - 2)^2 = 0 \]

### Step 2: Apply the difference of squares formula
The expression can be factored using the difference of squares:
\[ a^2 - b^2 = (a - b)(a + b) \]
Let \(a = (6x + 5)\) and \(b = (3x - 2)\). Thus, we have:
\[ (6x + 5 - (3x - 2))(6x + 5 + (3x - 2)) = 0 \]

### Step 3: Simplify the factors
Now, we simplify both factors:
1. For the first factor:
\[ 6x + 5 - 3x + 2 = 3x + 7 \]
2. For the second factor:
\[ 6x + 5 + 3x - 2 = 9x + 3 \]

So, we can rewrite the equation as:
\[ (3x + 7)(9x + 3) = 0 \]

### Step 4: Set each factor to zero
Now, we set each factor to zero:
1. \(3x + 7 = 0\)
2. \(9x + 3 = 0\)

### Step 5: Solve for \(x\)
1. From \(3x + 7 = 0\):
\[ 3x = -7 \implies x = -\frac{7}{3} \]
2. From \(9x + 3 = 0\):
\[ 9x = -3 \implies x = -\frac{1}{3} \]

### Step 6: Find the sum of the solutions
Now, we add the solutions:
\[ -\frac{7}{3} + -\frac{1}{3} = -\frac{8}{3} \]

### Final Answer
The sum of the solutions is:
\[ \boxed{-\frac{8}{3}} \]

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