`x^(2)-6x+8=0`
`{:("Quantity A","Quantity B"),(x^(2),2^(x)):}`

Options

A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the infromation given.

Answer

To solve the equation \( x^2 - 6x + 8 = 0 \) and compare the quantities \( A \) and \( B \), we will follow these steps:

### Step 1: Solve the quadratic equation
The given equation is:
\[ x^2 - 6x + 8 = 0 \]
We can use the quadratic formula to find the roots of the equation, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -6 \), and \( c = 8 \).

### Step 2: Substitute values into the quadratic formula
Substituting \( a \), \( b \), and \( c \) into the formula:
\[ x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{6 \pm \sqrt{36 - 32}}{2} \]

### Step 3: Simplify under the square root
Calculating the expression under the square root:
\[ x = \frac{6 \pm \sqrt{4}}{2} \]
Since \( \sqrt{4} = 2 \), we have:
\[ x = \frac{6 \pm 2}{2} \]

### Step 4: Calculate the two possible values of \( x \)
Now we can find the two values of \( x \):
1. \( x = \frac{6 + 2}{2} = \frac{8}{2} = 4 \)
2. \( x = \frac{6 - 2}{2} = \frac{4}{2} = 2 \)

### Step 5: Calculate Quantity A and Quantity B
Now we will calculate Quantity A and Quantity B for both values of \( x \).

**For \( x = 4 \):**
- Quantity A: \( A = x^2 = 4^2 = 16 \)
- Quantity B: \( B = 2^x = 2^4 = 16 \)

**For \( x = 2 \):**
- Quantity A: \( A = x^2 = 2^2 = 4 \)
- Quantity B: \( B = 2^x = 2^2 = 4 \)

### Step 6: Compare Quantity A and Quantity B
- For \( x = 4 \): \( A = 16 \) and \( B = 16 \) (both are equal)
- For \( x = 2 \): \( A = 4 \) and \( B = 4 \) (both are equal)

### Conclusion
In both cases, Quantity A and Quantity B are equal. Therefore, we conclude that:
\[ \text{Quantity A} = \text{Quantity B} \]

### Final Answer
The correct option is C: Quantity A is equal to Quantity B.

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