If `m^2-4m+1=0` then the value of `(m^3+1/m^3)` is:

Options

A. 48
B. 52
C. 64
D. none of these

Answer

To solve the equation \( m^2 - 4m + 1 = 0 \) and find the value of \( m^3 + \frac{1}{m^3} \), we can follow these steps:

### Step 1: Solve the quadratic equation
We start with the quadratic equation:
\[ m^2 - 4m + 1 = 0 \]
We can use the quadratic formula:
\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -4 \), and \( c = 1 \). Plugging in these values:
\[ m = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3} \]

### Step 2: Find \( m + \frac{1}{m} \)
Next, we need to find \( m + \frac{1}{m} \). We can express \( \frac{1}{m} \) in terms of \( m \):
\[ \frac{1}{m} = \frac{1}{2 \pm \sqrt{3}} \]
To rationalize this, we multiply the numerator and denominator by the conjugate:
\[ \frac{1}{m} = \frac{2 \mp \sqrt{3}}{(2 \pm \sqrt{3})(2 \mp \sqrt{3})} = \frac{2 \mp \sqrt{3}}{4 - 3} = 2 \mp \sqrt{3} \]
Thus, we have:
\[ m + \frac{1}{m} = (2 \pm \sqrt{3}) + (2 \mp \sqrt{3}) = 4 \]

### Step 3: Find \( m^3 + \frac{1}{m^3} \)
We can use the identity:
\[ m^3 + \frac{1}{m^3} = \left(m + \frac{1}{m}\right)^3 - 3\left(m + \frac{1}{m}\right) \]
Substituting \( m + \frac{1}{m} = 4 \):
\[ m^3 + \frac{1}{m^3} = 4^3 - 3 \cdot 4 \]
Calculating this:
\[ = 64 - 12 = 52 \]

### Final Answer
Thus, the value of \( m^3 + \frac{1}{m^3} \) is:
\[ \boxed{52} \]