If `5(3x+4)-8(6x+7)=9x-8`, then what is the value of `(x^2-2x+1)?`

Options

A. `2/3`
B. `4/9`
C. `5/3`
D. `25/9`

Answer

To solve the equation \( 5(3x + 4) - 8(6x + 7) = 9x - 8 \), we will follow these steps:

### Step 1: Distribute the terms on the left side
We start by distributing the \(5\) and \(-8\) across the parentheses:

\[ 5(3x + 4) = 15x + 20 \]
\[ -8(6x + 7) = -48x - 56 \]

### Step 2: Combine the distributed terms
Now we combine the results from Step 1:

\[ 15x + 20 - 48x - 56 = 9x - 8 \]

Combine like terms on the left side:

\[ (15x - 48x) + (20 - 56) = 9x - 8 \]
\[ -33x - 36 = 9x - 8 \]

### Step 3: Move all terms involving \(x\) to one side
Next, we will add \(33x\) to both sides of the equation:

\[ -36 = 9x + 33x - 8 \]
\[ -36 = 42x - 8 \]

### Step 4: Isolate the \(x\) term
Now, we will add \(8\) to both sides:

\[ -36 + 8 = 42x \]
\[ -28 = 42x \]

### Step 5: Solve for \(x\)
Now, we divide both sides by \(42\):

\[ x = \frac{-28}{42} \]

Simplifying this fraction gives:

\[ x = \frac{-2}{3} \]

### Step 6: Substitute \(x\) into the expression \(x^2 - 2x + 1\)
Now we need to find the value of \(x^2 - 2x + 1\):

\[ x^2 - 2x + 1 = \left(\frac{-2}{3}\right)^2 - 2\left(\frac{-2}{3}\right) + 1 \]

Calculating each term:

\[ \left(\frac{-2}{3}\right)^2 = \frac{4}{9} \]
\[ -2\left(\frac{-2}{3}\right) = \frac{4}{3} = \frac{12}{9} \]
\[ 1 = \frac{9}{9} \]

Now, combine these results:

\[ x^2 - 2x + 1 = \frac{4}{9} + \frac{12}{9} + \frac{9}{9} = \frac{4 + 12 + 9}{9} = \frac{25}{9} \]

### Final Answer
Thus, the value of \(x^2 - 2x + 1\) is:

\[ \frac{25}{9} \]

---