Solve the following quadratic: `\ x^2-4x+7=0`

Explanation

To solve the quadratic equation \( x^2 - 4x + 7 = 0 \), we will follow these steps:

### Step 1: Identify the coefficients
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). Here, we can identify:
- \( a = 1 \)
- \( b = -4 \)
- \( c = 7 \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-4)^2 - 4 \cdot 1 \cdot 7 \]
Calculating this:
\[ D = 16 - 28 = -12 \]

### Step 3: Analyze the discriminant
Since the discriminant \( D = -12 \) is less than zero, this indicates that the roots of the quadratic equation are imaginary.

### Step 4: Use the quadratic formula to find the roots
The roots of the quadratic equation can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values of \( b \), \( D \), and \( a \):
\[ x = \frac{-(-4) \pm \sqrt{-12}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{4 \pm \sqrt{-12}}{2} \]

### Step 5: Simplify the square root of the discriminant
The square root of \(-12\) can be simplified as follows:
\[ \sqrt{-12} = \sqrt{12} \cdot i = \sqrt{4 \cdot 3} \cdot i = 2\sqrt{3} \cdot i \]
Thus, we can rewrite the expression for \( x \):
\[ x = \frac{4 \pm 2\sqrt{3}i}{2} \]

### Step 6: Simplify the expression
Now we can simplify:
\[ x = 2 \pm \sqrt{3}i \]

### Final Result
The roots of the quadratic equation \( x^2 - 4x + 7 = 0 \) are:
\[ x = 2 + \sqrt{3}i \quad \text{and} \quad x = 2 - \sqrt{3}i \]