If x is rational and `4(x^(2)+(1)/(x^(2)))+16(x+(1)/(x))-57=0` , then the product of all possible values of x is

Options

A. 4
B. 3
C. 2
D. 1

Answer

To solve the equation \( 4\left(x^2 + \frac{1}{x^2}\right) + 16\left(x + \frac{1}{x}\right) - 57 = 0 \) and find the product of all possible rational values of \( x \), we can follow these steps:

### Step 1: Rewrite the equation
Start by letting \( y = x + \frac{1}{x} \). Then, we can express \( x^2 + \frac{1}{x^2} \) in terms of \( y \):
\[ x^2 + \frac{1}{x^2} = y^2 - 2 \]
Substituting this into the original equation gives:
\[ 4(y^2 - 2) + 16y - 57 = 0 \]

### Step 2: Simplify the equation
Now, simplify the equation:
\[ 4y^2 - 8 + 16y - 57 = 0 \]
Combine like terms:
\[ 4y^2 + 16y - 65 = 0 \]

### Step 3: Solve the quadratic equation
Now, we can use the quadratic formula to solve for \( y \):
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 4 \), \( b = 16 \), and \( c = -65 \):
\[ y = \frac{-16 \pm \sqrt{16^2 - 4 \cdot 4 \cdot (-65)}}{2 \cdot 4} \]
Calculate the discriminant:
\[ 16^2 = 256 \]
\[ -4 \cdot 4 \cdot (-65) = 1040 \]
Thus,
\[ b^2 - 4ac = 256 + 1040 = 1296 \]
Now, substituting back:
\[ y = \frac{-16 \pm \sqrt{1296}}{8} \]
Since \( \sqrt{1296} = 36 \):
\[ y = \frac{-16 \pm 36}{8} \]
This gives us two possible values for \( y \):
\[ y_1 = \frac{20}{8} = \frac{5}{2}, \quad y_2 = \frac{-52}{8} = -\frac{13}{2} \]

### Step 4: Find values of \( x \)
Now we need to find \( x \) from \( y \):
1. For \( y_1 = \frac{5}{2} \):
\[ x + \frac{1}{x} = \frac{5}{2} \]
Multiply through by \( 2x \):
\[ 2x^2 - 5x + 2 = 0 \]
Using the quadratic formula:
\[ x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot 2}}{2 \cdot 2} = \frac{5 \pm \sqrt{9}}{4} = \frac{5 \pm 3}{4} \]
This gives:
\[ x_1 = \frac{8}{4} = 2, \quad x_2 = \frac{2}{4} = \frac{1}{2} \]

2. For \( y_2 = -\frac{13}{2} \):
\[ x + \frac{1}{x} = -\frac{13}{2} \]
Multiply through by \( 2x \):
\[ 2x^2 + 13x + 2 = 0 \]
Using the quadratic formula:
\[ x = \frac{-13 \pm \sqrt{13^2 - 4 \cdot 2 \cdot 2}}{2 \cdot 2} = \frac{-13 \pm \sqrt{169 - 16}}{4} = \frac{-13 \pm \sqrt{153}}{4} \]
Since \( \sqrt{153} \) is not a rational number, this does not yield rational values for \( x \).

### Step 5: Calculate the product of all possible rational values of \( x \)
The rational solutions we found are \( x = 2 \) and \( x = \frac{1}{2} \). The product of these values is:
\[ 2 \cdot \frac{1}{2} = 1 \]

### Final Answer
The product of all possible rational values of \( x \) is \( \boxed{1} \).