To solve the quadratic equation:

\[
49x^2 - 14x + 1 = 0
\]

we can use one of the common methods for solving quadratic equations: factoring, completing the square, or using the quadratic formula. Since this equation represents a perfect square trinomial, it can be factored and solved easily.

### Step 1: Recognize the Perfect Square Trinomial

From the expression \(49x^2 - 14x + 1\), we noticed previously that it can be rewritten as:

\[
49x^2 - 14x + 1 = (7x - 1)^2
\]

### Step 2: Set the Factored Form Equal to Zero

Now, we set the factored form of the equation equal to zero:

\[
(7x - 1)^2 = 0
\]

### Step 3: Solve for \(x\)

Since \((7x - 1)^2 = 0\), this means:

\[
7x - 1 = 0
\]

Now solve for \(x\):

\[
7x = 1 \implies x = \frac{1}{7}
\]

### Solution

The only solution to the equation is:

\[
x = \frac{1}{7}
\]

### Verification

To ensure the solution is correct, substitute \(x = \frac{1}{7}\) back into the original equation:

\[
49\left(\frac{1}{7}\right)^2 - 14\left(\frac{1}{7}\right) + 1 = 49 \cdot \frac{1}{49} - 2 + 1 = 1 - 2 + 1 = 0
\]

Since the original equation equals zero with \(x = \frac{1}{7}\), the solution is verified.

Would you like more details or have any questions?

Here are some related questions to consider:

1. How would you solve \(49x^2 - 14x + 4 = 0\) using the quadratic formula?
2. Can you graph \(y = 49x^2 - 14x + 1\) and explain the vertex and axis of symmetry?
3. How does completing the square method work for solving quadratic equations?
4. What are some examples of quadratic equations that can't be factored easily?
5. How do you derive the quadratic formula from the general form of a quadratic equation?

**Tip:** Always try to factor the quadratic equation first, as it's often the quickest way to find the solution, especially if it's a perfect square trinomial.

Learn how to solve the quadratic equation 49x^2 - 14x + 1 = 0 with a detailed step-by-step solution. This problem involves factoring a perfect square trinomial and applying the quadratic formula. Suitable for students in Grades 9-12.

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