To find the value of \(x + \frac{1}{x}\) when \(x = 7 - 4\sqrt 3\), we need to first determine the value of \(\frac{1}{x}\) and then add it to \(x\).

Value of the Expression \(x + \frac{1}{x}\)

X Value: Deconstructing \(x = 7 - 4\sqrt 3\)

The problem provides us with the value of \(x\):

• \(x = 7 - 4\sqrt 3 \)

Our goal is to compute the expression \(x + \frac{1}{x}\). To do this, we first need to find the reciprocal of \(x\), which is \(\frac{1}{x}\).

Reciprocal Calculation: Determining \(\frac{1}{x}\)

To find \(\frac{1}{x}\), we substitute the given value of \(x\):

\[ \frac{1}{x} = \frac{1}{7 - 4\sqrt 3} \]

To simplify this expression and remove the square root from the denominator, we use the technique of rationalizing the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(7 - 4\sqrt 3\) is \(7 + 4\sqrt 3\).

We multiply the numerator and denominator by the conjugate:

\[ \frac{1}{x} = \frac{1}{7 - 4\sqrt 3} \times \frac{7 + 4\sqrt 3}{7 + 4\sqrt 3} \]

Now, we apply the difference of squares formula, \((a - b)(a + b) = a^2 - b^2\), in the denominator. Here, \(a = 7\) and \(b = 4\sqrt 3\).

\[ \frac{1}{x} = \frac{7 + 4\sqrt 3}{(7)^2 - (4\sqrt 3)^2} \]

Calculate the squares:

• \( (7)^2 = 49 \)
• \( (4\sqrt 3)^2 = 4^2 \times (\sqrt 3)^2 = 16 \times 3 = 48 \)

Substitute these values back into the expression:

\[ \frac{1}{x} = \frac{7 + 4\sqrt 3}{49 - 48} \]

\[ \frac{1}{x} = \frac{7 + 4\sqrt 3}{1} \]

So, the value of \(\frac{1}{x}\) is:

• \(\frac{1}{x} = 7 + 4\sqrt 3 \)

Expression Sum: Adding \(x\) and \(\frac{1}{x}\)

Now that we have both the original value of \(x\) and the calculated value of \(\frac{1}{x}\), we can find their sum for the expression \(x + \frac{1}{x}\):

\[ x + \frac{1}{x} = (7 - 4\sqrt 3) + (7 + 4\sqrt 3) \]

Combine the terms:

\[ x + \frac{1}{x} = 7 - 4\sqrt 3 + 7 + 4\sqrt 3 \]

Notice that the terms involving \(\sqrt 3\) are \( -4\sqrt 3 \) and \( +4\sqrt 3 \). These terms are additive inverses and will cancel each other out:

• \( -4\sqrt 3 + 4\sqrt 3 = 0 \)

So, the expression simplifies to:

\[ x + \frac{1}{x} = 7 + 7 \]

\[ x + \frac{1}{x} = 14 \]

Therefore, the final value of \(x + \frac{1}{x}\) is 14.

Comparing this result with the given options:

The calculated value matches Option 3.