Solve each of the following quadratic equations:
`100x^(2)-20x+1=0`

Explanation

To solve the quadratic equation \(100x^2 - 20x + 1 = 0\), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \(ax^2 + bx + c = 0\), where:
- \(a = 100\)
- \(b = -20\)
- \(c = 1\)

### Step 2: Rewrite the equation
We can observe that \(100x^2\) can be rewritten as \((10x)^2\) and \(1\) can be rewritten as \(1^2\). The equation can be expressed as:
\[ (10x)^2 - 2(10x)(1) + 1^2 = 0 \]

### Step 3: Recognize the perfect square trinomial
The expression \((10x)^2 - 2(10x)(1) + 1^2\) is a perfect square trinomial. Thus, we can factor it as:
\[ (10x - 1)^2 = 0 \]

### Step 4: Solve for \(x\)
To solve for \(x\), we set the factored expression equal to zero:
\[ 10x - 1 = 0 \]
Now, solving for \(x\):
\[ 10x = 1 \implies x = \frac{1}{10} \]

### Step 5: Identify the nature of the roots
Since we factored the quadratic into a perfect square, we find that the equation has real and equal roots. Therefore, the solution is:
\[ x = \frac{1}{10} \]

### Final Answer:
The roots of the equation \(100x^2 - 20x + 1 = 0\) are:
\[ x = \frac{1}{10} \quad \text{(with multiplicity 2)} \]

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