If ` x^(2) + x = 5` then the value of ` ( x + 3) ^(3) + (1)/( ( x + 3) ^(3) ) ` is

Options

A. 140
B. 110
C. 130
D. 120

Answer

To solve the equation \( x^2 + x = 5 \) and find the value of \( (x + 3)^3 + \frac{1}{(x + 3)^3} \), we can follow these steps:

### Step 1: Rearrange the equation
We start with the equation:
\[ x^2 + x - 5 = 0 \]

### Step 2: Solve for \( x \) using the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 1 \), and \( c = -5 \). Plugging in these values:
\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-5)}}{2 \cdot 1} \]
\[ x = \frac{-1 \pm \sqrt{1 + 20}}{2} \]
\[ x = \frac{-1 \pm \sqrt{21}}{2} \]

### Step 3: Define \( m \)
Let \( m = x + 3 \). Then:
\[ x = m - 3 \]

### Step 4: Substitute \( x \) in the original equation
Substituting \( x \) into the equation \( x^2 + x - 5 = 0 \):
\[ (m - 3)^2 + (m - 3) - 5 = 0 \]
Expanding this:
\[ (m^2 - 6m + 9) + (m - 3) - 5 = 0 \]
\[ m^2 - 6m + 9 + m - 3 - 5 = 0 \]
\[ m^2 - 5m + 1 = 0 \]

### Step 5: Solve for \( m \) using the quadratic formula
Using the quadratic formula again for \( m^2 - 5m + 1 = 0 \):
\[ m = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ m = \frac{5 \pm \sqrt{25 - 4}}{2} \]
\[ m = \frac{5 \pm \sqrt{21}}{2} \]

### Step 6: Calculate \( m^3 + \frac{1}{m^3} \)
To find \( m^3 + \frac{1}{m^3} \), we can use the identity:
\[ m^3 + \frac{1}{m^3} = \left(m + \frac{1}{m}\right)^3 - 3\left(m + \frac{1}{m}\right) \]
First, we need to calculate \( m + \frac{1}{m} \).

### Step 7: Calculate \( m + \frac{1}{m} \)
From \( m^2 - 5m + 1 = 0 \):
\[ m + \frac{1}{m} = 5 \]

### Step 8: Substitute into the identity
Now substitute \( m + \frac{1}{m} \) into the identity:
\[ m^3 + \frac{1}{m^3} = 5^3 - 3 \cdot 5 \]
\[ = 125 - 15 \]
\[ = 110 \]

### Final Answer
Thus, the value of \( (x + 3)^3 + \frac{1}{(x + 3)^3} \) is:
\[ \boxed{110} \]