If both expression `x^2 - 3x + 2 and x^2 - 4x + 3` equal 0. then what is the value of `(x - 3)^2` ?

Options

A. 0
B. 1
C. 2
D. 4

Answer

To solve the problem, we need to find the value of \((x - 3)^2\) given that both expressions \(x^2 - 3x + 2 = 0\) and \(x^2 - 4x + 3 = 0\).

### Step-by-Step Solution:

1. **Solve the first equation: \(x^2 - 3x + 2 = 0\)**

We can factor this quadratic equation:
\[ x^2 - 3x + 2 = (x - 1)(x - 2) = 0 \]
Setting each factor to zero gives us:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
So, the solutions for the first equation are \(x = 1\) and \(x = 2\).

2. **Solve the second equation: \(x^2 - 4x + 3 = 0\)**

We can factor this quadratic equation as well:
\[ x^2 - 4x + 3 = (x - 1)(x - 3) = 0 \]
Setting each factor to zero gives us:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]
So, the solutions for the second equation are \(x = 1\) and \(x = 3\).

3. **Find the common solution:**

From the first equation, we have the solutions \(x = 1\) and \(x = 2\).
From the second equation, we have the solutions \(x = 1\) and \(x = 3\).
The common solution is:
\[ x = 1 \]

4. **Calculate \((x - 3)^2\):**

Now that we have \(x = 1\), we can substitute this value into \((x - 3)^2\):
\[ (x - 3)^2 = (1 - 3)^2 = (-2)^2 = 4 \]

Thus, the value of \((x - 3)^2\) is **4**.