SOLVE THE QUADRATIC EQUATION BY COMPLETING THE SQUARE x^2 - 4x + 1 = 0

To solve the quadratic equation \(x^2 - 4x + 1 = 0\) by completing the square, follow these steps:

### Step 1: Move the constant term to the other side of the equation.
\[
x^2 - 4x = -1
\]

### Step 2: Complete the square.
To complete the square, we need to add and subtract the square of half the coefficient of \(x\) on the left side. The coefficient of \(x\) is \(-4\), so half of it is \(-2\), and squaring it gives \(4\).

Add and subtract 4 to the left side:
\[
x^2 - 4x + 4 = -1 + 4
\]

### Step 3: Simplify the equation.
Now the left side is a perfect square trinomial, and the right side simplifies:
\[
(x - 2)^2 = 3
\]

### Step 4: Take the square root of both sides.
\[
x - 2 = \pm \sqrt{3}
\]

### Step 5: Solve for \(x\).
Finally, isolate \(x\) by adding 2 to both sides:
\[
x = 2 \pm \sqrt{3}
\]

So the solutions are:
\[
x = 2 + \sqrt{3} \quad \text{or} \quad x = 2 - \sqrt{3}
\]

Would you like more details or have any questions? Here are some related questions to explore:

1. How do you derive the quadratic formula?
2. Can you solve a quadratic equation by factoring?
3. What is the significance of the discriminant in a quadratic equation?
4. How does the graph of a quadratic function relate to its equation?
5. What are real-world applications of solving quadratic equations?

**Tip:** Completing the square is also useful for deriving the quadratic formula and analyzing the vertex form of a quadratic function.

Learn how to solve the quadratic equation x^2 - 4x + 1 = 0 step-by-step using the method of completing the square. This problem involves key algebraic concepts such as quadratic equations, completing the square, and the quadratic formula. Suitable for high school students in Grades 9-12.

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