If `x^(2)-4x+1=0`, then what is the value of `x^(9) + x^(7) -194x^(5)-194x^(3)` ?

Options

A. 4
B. `-4`
C. 1
D. `-1`

Answer

To solve the equation \( x^2 - 4x + 1 = 0 \) and find the value of \( x^9 + x^7 - 194x^5 - 194x^3 \), we can follow these steps:

### Step 1: Solve the quadratic equation
We start with the equation:
\[ x^2 - 4x + 1 = 0 \]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -4, c = 1 \):
\[ x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3} \]
Thus, the roots are:
\[ x_1 = 2 + \sqrt{3}, \quad x_2 = 2 - \sqrt{3} \]

### Step 2: Find \( x^2 \)
From the original equation, we can express \( x^2 \) in terms of \( x \):
\[ x^2 = 4x - 1 \]

### Step 3: Calculate higher powers of \( x \)
Using \( x^2 = 4x - 1 \), we can find \( x^3 \):
\[ x^3 = x \cdot x^2 = x(4x - 1) = 4x^2 - x = 4(4x - 1) - x = 16x - 4 - x = 15x - 4 \]

Now, calculate \( x^4 \):
\[ x^4 = x \cdot x^3 = x(15x - 4) = 15x^2 - 4x = 15(4x - 1) - 4x = 60x - 15 - 4x = 56x - 15 \]

Next, calculate \( x^5 \):
\[ x^5 = x \cdot x^4 = x(56x - 15) = 56x^2 - 15x = 56(4x - 1) - 15x = 224x - 56 - 15x = 209x - 56 \]

Now, calculate \( x^6 \):
\[ x^6 = x \cdot x^5 = x(209x - 56) = 209x^2 - 56x = 209(4x - 1) - 56x = 836x - 209 - 56x = 780x - 209 \]

Next, calculate \( x^7 \):
\[ x^7 = x \cdot x^6 = x(780x - 209) = 780x^2 - 209x = 780(4x - 1) - 209x = 3120x - 780 - 209x = 2911x - 780 \]

Next, calculate \( x^8 \):
\[ x^8 = x \cdot x^7 = x(2911x - 780) = 2911x^2 - 780x = 2911(4x - 1) - 780x = 11644x - 2911 - 780x = 10864x - 2911 \]

Finally, calculate \( x^9 \):
\[ x^9 = x \cdot x^8 = x(10864x - 2911) = 10864x^2 - 2911x = 10864(4x - 1) - 2911x = 43456x - 10864 - 2911x = 40545x - 10864 \]

### Step 4: Substitute into the expression
Now we substitute \( x^9 \) and \( x^7 \) into the expression \( x^9 + x^7 - 194x^5 - 194x^3 \):
\[ x^9 + x^7 - 194x^5 - 194x^3 = (40545x - 10864) + (2911x - 780) - 194(209x - 56) - 194(15x - 4) \]
Calculating \( -194(209x - 56) \) and \( -194(15x - 4) \):
\[ -194(209x - 56) = -40546x + 10864 \]
\[ -194(15x - 4) = -2910x + 776 \]

Putting it all together:
\[ (40545x - 10864) + (2911x - 780) - 40546x + 10864 - 2910x + 776 \]
Combining like terms:
\[ (40545x + 2911x - 40546x - 2910x) + (-10864 - 780 + 10864 + 776) = 0x + (-780 + 776) = -4 \]

Thus, the final answer is:
\[ \boxed{-4} \]