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

Options

A. 44
B. 48
C. 52
D. 64

Answer

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

### Step 1: Solve the quadratic equation
We start with the equation:
\[ t^2 - 4t + 1 = 0 \]
We can use the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -4 \), and \( c = 1 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot 1 = 16 - 4 = 12 \]
Now substituting into the quadratic formula:
\[ t = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3} \]
Thus, the two possible values for \( t \) are:
\[ t_1 = 2 + \sqrt{3} \quad \text{and} \quad t_2 = 2 - \sqrt{3} \]

### Step 2: Find \( t^3 \)
We can express \( t^3 \) in terms of \( t \) using the original equation. From \( t^2 = 4t - 1 \), we can multiply both sides by \( t \):
\[ t^3 = t \cdot t^2 = t(4t - 1) = 4t^2 - t \]
Now substituting \( t^2 \) back in:
\[ t^3 = 4(4t - 1) - t = 16t - 4 - t = 15t - 4 \]

### Step 3: Find \( t^3 + \frac{1}{t^3} \)
To find \( \frac{1}{t^3} \), we can use the identity:
\[ \frac{1}{t^3} = \frac{1}{15t - 4} \]
Now we need to compute \( t^3 + \frac{1}{t^3} \):
\[ t^3 + \frac{1}{t^3} = (15t - 4) + \frac{1}{15t - 4} \]
To simplify this expression, we can find a common denominator:
\[ = \frac{(15t - 4)^2 + 1}{15t - 4} \]
Calculating \( (15t - 4)^2 + 1 \):
\[ (15t - 4)^2 = 225t^2 - 120t + 16 \]
Since \( t^2 = 4t - 1 \):
\[ 225(4t - 1) - 120t + 16 = 900t - 225 - 120t + 16 = 780t - 209 \]
Thus,
\[ t^3 + \frac{1}{t^3} = \frac{780t - 209 + 1}{15t - 4} = \frac{780t - 208}{15t - 4} \]

### Step 4: Substitute \( t = 2 + \sqrt{3} \) or \( t = 2 - \sqrt{3} \)
We can substitute either value of \( t \) into our expression. Let's use \( t = 2 + \sqrt{3} \):
\[ t^3 = 15(2 + \sqrt{3}) - 4 = 30 + 15\sqrt{3} - 4 = 26 + 15\sqrt{3} \]
Now, we can find \( \frac{1}{t^3} \):
\[ \frac{1}{t^3} = \frac{1}{26 + 15\sqrt{3}} \]
To rationalize the denominator:
\[ \frac{1}{t^3} = \frac{26 - 15\sqrt{3}}{(26 + 15\sqrt{3})(26 - 15\sqrt{3})} = \frac{26 - 15\sqrt{3}}{676 - 675} = 26 - 15\sqrt{3} \]

### Step 5: Combine \( t^3 \) and \( \frac{1}{t^3} \)
Now we can add:
\[ t^3 + \frac{1}{t^3} = (26 + 15\sqrt{3}) + (26 - 15\sqrt{3}) = 52 \]

Thus, the final answer is:
\[ \boxed{52} \]