If `(3x + 4)^2 + (3x - 2)^2 = (6x + 5)(3x - 2) + 12` , then the value of x is:

Options

A. `2`
B. `1`
C. `-3`
D. `-2`

Answer

To solve the equation \((3x + 4)^2 + (3x - 2)^2 = (6x + 5)(3x - 2) + 12\), we will follow these steps:

### Step 1: Expand both sides of the equation
We will expand the left-hand side (LHS) and the right-hand side (RHS) of the equation.

**LHS:**
\[ (3x + 4)^2 + (3x - 2)^2 \]
Using the formula \((a + b)^2 = a^2 + 2ab + b^2\) and \((a - b)^2 = a^2 - 2ab + b^2\):

1. Expand \((3x + 4)^2\):
\[ (3x)^2 + 2(3x)(4) + (4)^2 = 9x^2 + 24x + 16 \]

2. Expand \((3x - 2)^2\):
\[ (3x)^2 - 2(3x)(2) + (2)^2 = 9x^2 - 12x + 4 \]

Now, combine these:
\[ LHS = (9x^2 + 24x + 16) + (9x^2 - 12x + 4) = 18x^2 + 12x + 20 \]

**RHS:**
\[ (6x + 5)(3x - 2) + 12 \]
Using the distributive property:
1. Multiply \(6x\) with both terms in \((3x - 2)\):
\[ 6x \cdot 3x - 6x \cdot 2 = 18x^2 - 12x \]

2. Multiply \(5\) with both terms in \((3x - 2)\):
\[ 5 \cdot 3x - 5 \cdot 2 = 15x - 10 \]

Now combine these:
\[ RHS = (18x^2 - 12x + 15x - 10) + 12 = 18x^2 + 3x + 2 \]

### Step 2: Set the LHS equal to the RHS
Now we have:
\[ 18x^2 + 12x + 20 = 18x^2 + 3x + 2 \]

### Step 3: Simplify the equation
Subtract \(18x^2\) from both sides:
\[ 12x + 20 = 3x + 2 \]

Now, subtract \(3x\) from both sides:
\[ 12x - 3x + 20 = 2 \]
This simplifies to:
\[ 9x + 20 = 2 \]

### Step 4: Isolate \(x\)
Subtract \(20\) from both sides:
\[ 9x = 2 - 20 \]
\[ 9x = -18 \]

Now, divide both sides by \(9\):
\[ x = \frac{-18}{9} = -2 \]

### Final Answer
The value of \(x\) is \(-2\).

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