If `x=(8+-sqrt((-8)^2-4xx3xx2))/(2xx3)` then the required polynomial is:

Options

A. `3x^2-8x+2=0`
B. `2x^2-8x-2=0`
C. `3x^2+8x-2=0`
D. `3x^2+8x+2=0`

Answer

To find the required polynomial from the given expression for \( x \), we will follow these steps:

### Step 1: Write down the expression for \( x \)
Given:
\[ x = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3} \]

### Step 2: Simplify the expression under the square root
Calculate \( (-8)^2 \):
\[ (-8)^2 = 64 \]
Now calculate \( 4 \cdot 3 \cdot 2 \):
\[ 4 \cdot 3 \cdot 2 = 24 \]
Now substitute these values back into the expression:
\[ x = \frac{8 \pm \sqrt{64 - 24}}{6} \]

### Step 3: Simplify the square root
Calculate \( 64 - 24 \):
\[ 64 - 24 = 40 \]
Now substitute this back:
\[ x = \frac{8 \pm \sqrt{40}}{6} \]

### Step 4: Simplify \( \sqrt{40} \)
We can simplify \( \sqrt{40} \):
\[ \sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10} \]
Now substitute this back into the equation:
\[ x = \frac{8 \pm 2\sqrt{10}}{6} \]

### Step 5: Simplify the fraction
Divide each term in the numerator by 6:
\[ x = \frac{8}{6} \pm \frac{2\sqrt{10}}{6} = \frac{4}{3} \pm \frac{\sqrt{10}}{3} \]
Thus, we can rewrite \( x \) as:
\[ x = \frac{4 \pm \sqrt{10}}{3} \]

### Step 6: Identify coefficients for the polynomial
From the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
We can compare:
- \( -b = 4 \) implies \( b = -4 \)
- \( 2a = 3 \) implies \( a = \frac{3}{2} \)
- \( c = 2 \)

### Step 7: Write the polynomial
Using \( a = 3 \), \( b = -8 \), and \( c = 2 \), we can write the polynomial:
\[ 3x^2 - 8x + 2 = 0 \]

### Final Answer
The required polynomial is:
\[ 3x^2 - 8x + 2 = 0 \]

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