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 given the expression for \( x \), we can follow these steps:

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

### Step 2: Simplify the expression under the square root
Calculate \( (-8)^2 - 4 \cdot 3 \cdot 2 \):
\[ (-8)^2 = 64 \]
\[ 4 \cdot 3 \cdot 2 = 24 \]
Now, substitute these values:
\[ 64 - 24 = 40 \]
So, the expression for \( x \) becomes:
\[ x = \frac{8 \pm \sqrt{40}}{6} \]

### Step 3: Simplify \( \sqrt{40} \)
We can simplify \( \sqrt{40} \):
\[ \sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10} \]
Thus, the expression for \( x \) now is:
\[ x = \frac{8 \pm 2\sqrt{10}}{6} \]

### Step 4: Further simplify the expression for \( x \)
We can simplify this fraction:
\[ x = \frac{8}{6} \pm \frac{2\sqrt{10}}{6} \]
\[ x = \frac{4}{3} \pm \frac{\sqrt{10}}{3} \]
This means:
\[ x = \frac{4 \pm \sqrt{10}}{3} \]

### Step 5: Set up the standard form of the quadratic polynomial
We know that the roots of a quadratic polynomial can be expressed in the form:
\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
From our expression, we can identify:
- \( A = 3 \)
- \( B = -8 \)
- \( C = 2 \)

### Step 6: Write the polynomial
Using the standard form \( Ax^2 + Bx + C = 0 \), we substitute the values of \( A \), \( B \), and \( C \):
\[ 3x^2 - 8x + 2 = 0 \]
Thus, the required polynomial is:
\[ 3x^2 - 8x + 2 \]

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

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