Let's solve the problem step-by-step.
### Step 1: Solve the Quadratic Equation
The given quadratic equation is:
[tex]\[ x^2 + 4x - 1 = 0 \][/tex]
We can solve for [tex]\( x \)[/tex] using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -1 \)[/tex].
Substitute these values into the formula:
[tex]\[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-4 \pm \sqrt{16 + 4}}{2} \][/tex]
[tex]\[ x = \frac{-4 \pm \sqrt{20}}{2} \][/tex]
[tex]\[ x = \frac{-4 \pm 2\sqrt{5}}{2} \][/tex]
[tex]\[ x = -2 \pm \sqrt{5} \][/tex]
Since [tex]\( x \)[/tex] is positive, we'll take the positive root:
[tex]\[ x = -2 + \sqrt{5} \][/tex]
### Step 2: Calculate [tex]\( x + \frac{1}{x} \)[/tex]
First, let's find [tex]\( \frac{1}{x} \)[/tex]. Given [tex]\( x = -2 + \sqrt{5} \)[/tex], we need to rationalize it:
[tex]\[ \frac{1}{x} = \frac{1}{-2 + \sqrt{5}} \][/tex]
To rationalize, multiply by the conjugate:
[tex]\[ \frac{1}{x} = \frac{1}{-2 + \sqrt{5}} \cdot \frac{-2 - \sqrt{5}}{-2 - \sqrt{5}} = \frac{-2 - \sqrt{5}}{4 - 5} = \frac{-2 - \sqrt{5}}{-1} = 2 + \sqrt{5} \][/tex]
Now, let's add [tex]\( x \)[/tex] to [tex]\( \frac{1}{x} \)[/tex]:
[tex]\[ x + \frac{1}{x} = (-2 + \sqrt{5}) + (2 + \sqrt{5}) = \sqrt{5} + \sqrt{5} = 2\sqrt{5} \][/tex]
### Step 3: Calculate [tex]\( x^2 + \frac{1}{x^2} \)[/tex]
We need to find [tex]\( x^2 \)[/tex] and [tex]\( \frac{1}{x^2} \)[/tex].
First, compute [tex]\( x^2 \)[/tex]:
[tex]\[ x^2 = (-2 + \sqrt{5})^2 = 4 - 4\sqrt{5} + 5 = 9 - 4\sqrt{5} \][/tex]
Next, compute [tex]\( \frac{1}{x^2} \)[/tex]:
[tex]\[ x = -2 + \sqrt{5} \Rightarrow \frac{1}{x} = 2 + \sqrt{5} \][/tex]
[tex]\[ \frac{1}{x^2} = (2 + \sqrt{5})^2 = 4 + 4\sqrt{5} + 5 = 9 + 4\sqrt{5} \][/tex]
Now add [tex]\( x^2 \)[/tex] and [tex]\( \frac{1}{x^2} \)[/tex]:
[tex]\[ x^2 + \frac{1}{x^2} = (9 - 4\sqrt{5}) + (9 + 4\sqrt{5}) = 9 + 9 = 18 \][/tex]
### Summary of Results
(a) [tex]\( x + \frac{1}{x} = 2\sqrt{5} \)[/tex]
(b) [tex]\( x^2 + \frac{1}{x^2} = 18 \)[/tex]
