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 equation:
\[ m^2 - 4m + 1 = 0 \]
We can use the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -4, c = 1 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot 1 = 16 - 4 = 12 \]
Now substituting into the quadratic formula:
\[ m = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3} \]
Thus, the roots are:
\[ m_1 = 2 + \sqrt{3} \quad \text{and} \quad m_2 = 2 - \sqrt{3} \]

### Step 2: Find \( m + \frac{1}{m} \)
Using the relationship derived from the quadratic equation, we can express \( m + \frac{1}{m} \):
\[ m + \frac{1}{m} = 4 \]
This is derived from dividing the original equation by \( m \).

### 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 \( 4^3 \):
\[ 4^3 = 64 \]
Now substituting back:
\[ m^3 + \frac{1}{m^3} = 64 - 12 = 52 \]

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