Solve for `x : 6x^(2) + 11x + 3 = 0`

Answer

To solve the quadratic equation \( 6x^2 + 11x + 3 = 0 \), we can use the method of splitting the middle term. Here’s a step-by-step solution:

### Step 1: Identify the coefficients
The quadratic equation is in the standard form \( ax^2 + bx + c = 0 \).
- Here, \( a = 6 \), \( b = 11 \), and \( c = 3 \).

### Step 2: Calculate the product \( ac \)
We need to find the product of \( a \) and \( c \):
\[ ac = 6 \times 3 = 18 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( 18 \) (the value of \( ac \)) and add up to \( 11 \) (the value of \( b \)). The numbers are \( 9 \) and \( 2 \) because:
\[ 9 \times 2 = 18 \quad \text{and} \quad 9 + 2 = 11 \]

### Step 4: Rewrite the middle term
Now we can rewrite the equation by splitting the middle term using the numbers we found:
\[ 6x^2 + 9x + 2x + 3 = 0 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (6x^2 + 9x) + (2x + 3) = 0 \]

### Step 6: Factor by grouping
Now we factor out the common factors from each group:
\[ 3x(2x + 3) + 1(2x + 3) = 0 \]

### Step 7: Factor out the common binomial
We can now factor out the common binomial \( (2x + 3) \):
\[ (2x + 3)(3x + 1) = 0 \]

### Step 8: Set each factor to zero
Now we set each factor equal to zero:
1. \( 2x + 3 = 0 \)
2. \( 3x + 1 = 0 \)

### Step 9: Solve for \( x \)
For the first equation:
\[ 2x + 3 = 0 \implies 2x = -3 \implies x = -\frac{3}{2} \]

For the second equation:
\[ 3x + 1 = 0 \implies 3x = -1 \implies x = -\frac{1}{3} \]

### Final Solution
The solutions for \( x \) are:
\[ x = -\frac{3}{2} \quad \text{and} \quad x = -\frac{1}{3} \]

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