x²+6x+4=0 (solve by completing the square)

To solve the quadratic equation \(x^2 + 6x + 4 = 0\) by completing the square, follow these steps:

### Step 1: Move the constant term to the right side of the equation.
\[
x^2 + 6x = -4
\]

### Step 2: Add the square of half the coefficient of \(x\) to both sides.
The coefficient of \(x\) is 6. Half of 6 is 3, and squaring 3 gives \(3^2 = 9\).

Add 9 to both sides:
\[
x^2 + 6x + 9 = -4 + 9
\]

Simplifying the right side:
\[
x^2 + 6x + 9 = 5
\]

### Step 3: Write the left side as a perfect square.
The left side can be written as:
\[
(x + 3)^2 = 5
\]

### Step 4: Solve for \(x\) by taking the square root of both sides.
\[
x + 3 = \pm \sqrt{5}
\]

### Step 5: Isolate \(x\).
\[
x = -3 \pm \sqrt{5}
\]

So, the solutions are:
\[
x = -3 + \sqrt{5} \quad \text{or} \quad x = -3 - \sqrt{5}
\]

### Final Answer:
\[
x = -3 + \sqrt{5}, \quad x = -3 - \sqrt{5}
\]

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

1. How would you solve the equation \(x^2 + 6x + 9 = 0\) using completing the square?
2. What is the general formula for solving any quadratic equation by completing the square?
3. Can you explain how the discriminant relates to the solutions found by completing the square?
4. What are the advantages and disadvantages of completing the square versus using the quadratic formula?
5. How would you solve the equation \(2x^2 + 8x + 3 = 0\) by completing the square?

**Tip:** Completing the square is especially useful for solving quadratic equations when you want to derive the vertex form of a parabola.

Learn how to solve the quadratic equation x^2 + 6x + 4 = 0 using the method of completing the square. Follow detailed step-by-step instructions and understand the algebraic concepts involved. Suitable for advanced high school students (Grades 10-12).

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