If ` x ( x - 3) = - 1 ` then the value of ` x^(3) ( x^(3) - 18)` is

Options

A. `-1`
B. 2
C. 1
D. 0

Answer

To solve the equation \( x (x - 3) = -1 \) and find the value of \( x^3 (x^3 - 18) \), we can follow these steps:

### Step 1: Rewrite the equation
Starting from the equation:
\[ x (x - 3) = -1 \]
We can rewrite it as:
\[ x^2 - 3x + 1 = 0 \]

### Step 2: Solve the quadratic equation
To solve for \( x \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = -3 \), and \( c = 1 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-3)^2 - 4(1)(1) = 9 - 4 = 5 \]
Now substituting into the quadratic formula:
\[ x = \frac{3 \pm \sqrt{5}}{2} \]

### Step 3: Calculate \( x^3 \)
Next, we need to find \( x^3 \). We can use the identity \( x^3 = x(x^2) \). From our quadratic equation, we know:
\[ x^2 = 3x - 1 \]
Thus,
\[ x^3 = x(3x - 1) = 3x^2 - x \]
Substituting \( x^2 \) back into the equation:
\[ x^3 = 3(3x - 1) - x = 9x - 3 - x = 8x - 3 \]

### Step 4: Find \( x^3 - 18 \)
Now we need to calculate \( x^3 - 18 \):
\[ x^3 - 18 = (8x - 3) - 18 = 8x - 21 \]

### Step 5: Calculate \( x^3 (x^3 - 18) \)
Now we can find \( x^3 (x^3 - 18) \):
\[ x^3 (x^3 - 18) = (8x - 3)(8x - 21) \]
Expanding this:
\[ = 64x^2 - 168x + 63 \]

### Step 6: Substitute \( x^2 \)
We know \( x^2 = 3x - 1 \), so substituting \( x^2 \) into the expression:
\[ = 64(3x - 1) - 168x + 63 \]
\[ = 192x - 64 - 168x + 63 \]
\[ = 24x - 1 \]

### Step 7: Substitute \( x \)
Now we can substitute \( x = \frac{3 \pm \sqrt{5}}{2} \) into \( 24x - 1 \):
\[ = 24\left(\frac{3 \pm \sqrt{5}}{2}\right) - 1 = 12(3 \pm \sqrt{5}) - 1 = 36 \pm 12\sqrt{5} - 1 \]
\[ = 35 \pm 12\sqrt{5} \]

### Final Answer
Thus, the value of \( x^3 (x^3 - 18) \) is \( 35 \pm 12\sqrt{5} \).