Find the roots of the equation `x^2 -10x+25=0`

Answer

To find the roots of the equation \( x^2 - 10x + 25 = 0 \), we can use the method of factorization.

### Step-by-step Solution:

1. **Identify the quadratic equation**:
The given equation is \( x^2 - 10x + 25 = 0 \).

2. **Rewrite the equation**:
We can observe that the equation can be factored. We look for two numbers that multiply to \( 25 \) (the constant term) and add up to \( -10 \) (the coefficient of \( x \)). The numbers \( -5 \) and \( -5 \) satisfy these conditions.

3. **Factor the quadratic**:
We can rewrite the equation as:
\[ x^2 - 5x - 5x + 25 = 0 \]
This can be grouped as:
\[ (x^2 - 5x) + (-5x + 25) = 0 \]
Factoring by grouping gives:
\[ x(x - 5) - 5(x - 5) = 0 \]

4. **Factor out the common term**:
Now we can factor out \( (x - 5) \):
\[ (x - 5)(x - 5) = 0 \]
This can also be written as:
\[ (x - 5)^2 = 0 \]

5. **Solve for \( x \)**:
Setting the factor equal to zero gives:
\[ x - 5 = 0 \]
Thus:
\[ x = 5 \]

6. **Identify the roots**:
Since the factor \( (x - 5) \) appears twice, the root \( x = 5 \) has a multiplicity of 2. Therefore, the roots of the equation are:
\[ x = 5, 5 \]

### Final Answer:
The roots of the equation \( x^2 - 10x + 25 = 0 \) are \( x = 5 \) (with multiplicity 2).