Find the roots of the equations.
Q. `x^(2)-2x+5=0`

Answer

To find the roots of the quadratic equation \(x^2 - 2x + 5 = 0\), we can follow these steps:

### Step 1: Identify coefficients
The given quadratic equation is in the standard form \(ax^2 + bx + c = 0\). Here, we identify:
- \(a = 1\)
- \(b = -2\)
- \(c = 5\)

### Step 2: Calculate the discriminant
The discriminant \(D\) is calculated using the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ D = (-2)^2 - 4 \cdot 1 \cdot 5 \]
\[ D = 4 - 20 \]
\[ D = -16 \]

### Step 3: Determine the nature of the roots
Since the discriminant \(D\) is negative (\(-16\)), it indicates that the roots are complex.

### Step 4: Use the quadratic formula to find the roots
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values of \(b\), \(D\), and \(a\):
\[ x = \frac{-(-2) \pm \sqrt{-16}}{2 \cdot 1} \]
\[ x = \frac{2 \pm \sqrt{-16}}{2} \]

### Step 5: Simplify the square root
The square root of \(-16\) can be expressed using imaginary numbers:
\[ \sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i \]
where \(i\) is the imaginary unit.

### Step 6: Substitute back into the formula
Now substituting back:
\[ x = \frac{2 \pm 4i}{2} \]

### Step 7: Simplify the expression
This gives us two roots:
\[ x = \frac{2 + 4i}{2} = 1 + 2i \]
\[ x = \frac{2 - 4i}{2} = 1 - 2i \]

### Final Answer
Thus, the roots of the equation \(x^2 - 2x + 5 = 0\) are:
\[ x = 1 + 2i \quad \text{and} \quad x = 1 - 2i \]

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