Solve the following quadratic equation by factorization method only: `x^2-x+1=0`

Explanation

To solve the quadratic equation \( x^2 - x + 1 = 0 \) using the factorization method, we will follow these steps:

### Step 1: Write the equation in standard form
The given equation is already in standard form:
\[ x^2 - x + 1 = 0 \]

### Step 2: Identify coefficients
In the equation \( ax^2 + bx + c = 0 \), we have:
- \( a = 1 \)
- \( b = -1 \)
- \( c = 1 \)

### Step 3: Calculate the discriminant
The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-1)^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \]

### Step 4: Analyze the discriminant
Since the discriminant \( D \) is negative (\( D < 0 \)), this indicates that the quadratic equation has complex roots.

### Step 5: Write the equation in a form suitable for completing the square
We can rewrite the equation as:
\[ x^2 - x + 1 = 0 \]
To complete the square, we can rearrange it:
\[ x^2 - x = -1 \]

### Step 6: Complete the square
To complete the square, we take half of the coefficient of \( x \) (which is \(-1\)), square it, and add it to both sides:
\[ \left( \frac{-1}{2} \right)^2 = \frac{1}{4} \]
Now add \( \frac{1}{4} \) to both sides:
\[ x^2 - x + \frac{1}{4} = -1 + \frac{1}{4} \]
This simplifies to:
\[ x^2 - x + \frac{1}{4} = -\frac{3}{4} \]

### Step 7: Factor the left-hand side
The left-hand side can be factored as:
\[ \left( x - \frac{1}{2} \right)^2 = -\frac{3}{4} \]

### Step 8: Take the square root of both sides
Taking the square root of both sides gives:
\[ x - \frac{1}{2} = \pm \sqrt{-\frac{3}{4}} \]
This can be rewritten using \( i \) (the imaginary unit):
\[ x - \frac{1}{2} = \pm \frac{\sqrt{3}}{2} i \]

### Step 9: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{1}{2} \pm \frac{\sqrt{3}}{2} i \]

### Final Answer
Thus, the solutions to the quadratic equation \( x^2 - x + 1 = 0 \) are:
\[ x = \frac{1}{2} + \frac{\sqrt{3}}{2} i \quad \text{and} \quad x = \frac{1}{2} - \frac{\sqrt{3}}{2} i \]