Solve: `2x^(2) - 4x + 3 = 0`

Answer

To solve the quadratic equation \( 2x^2 - 4x + 3 = 0 \), we will use the quadratic formula, which is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 1: Identify coefficients
In the equation \( 2x^2 - 4x + 3 = 0 \), we identify the coefficients:
- \( a = 2 \)
- \( b = -4 \)
- \( c = 3 \)

### Step 2: Calculate the discriminant
Next, we calculate the discriminant \( D = b^2 - 4ac \):

\[ D = (-4)^2 - 4 \cdot 2 \cdot 3 \]
\[ D = 16 - 24 \]
\[ D = -8 \]

### Step 3: Substitute into the quadratic formula
Since the discriminant is negative, we will have complex solutions. Now, we substitute \( a \), \( b \), and \( D \) into the quadratic formula:

\[ x = \frac{-(-4) \pm \sqrt{-8}}{2 \cdot 2} \]
\[ x = \frac{4 \pm \sqrt{-8}}{4} \]

### Step 4: Simplify the square root
We simplify \( \sqrt{-8} \):

\[ \sqrt{-8} = \sqrt{8} \cdot i = \sqrt{4 \cdot 2} \cdot i = 2\sqrt{2}i \]

### Step 5: Substitute back into the formula
Now, substituting back into the equation gives:

\[ x = \frac{4 \pm 2\sqrt{2}i}{4} \]

### Step 6: Simplify the expression
We can simplify this expression:

\[ x = \frac{4}{4} \pm \frac{2\sqrt{2}i}{4} \]
\[ x = 1 \pm \frac{\sqrt{2}}{2}i \]

### Final Answer
Thus, the solutions for the equation \( 2x^2 - 4x + 3 = 0 \) are:

\[ x = 1 + \frac{\sqrt{2}}{2}i \quad \text{and} \quad x = 1 - \frac{\sqrt{2}}{2}i \]

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