solving quadratic equations using completing squares
x^2 - 4x + 12 = 0

To solve the quadratic equation \(x^2 - 4x + 12 = 0\) using the method of completing the square, follow these steps:

### Step 1: Move the constant term to the right side of the equation
\[
x^2 - 4x = -12
\]

### Step 2: Complete the square
To complete the square, take half of the coefficient of \(x\), square it, and add it to both sides of the equation.

The coefficient of \(x\) is \(-4\). Half of \(-4\) is \(-2\), and \((-2)^2 = 4\).

Add 4 to both sides:
\[
x^2 - 4x + 4 = -12 + 4
\]
\[
(x - 2)^2 = -8
\]

### Step 3: Solve for \(x\)
Now, take the square root of both sides:
\[
x - 2 = \pm \sqrt{-8}
\]

Since \(\sqrt{-8}\) involves the imaginary unit \(i\), rewrite it as:
\[
x - 2 = \pm \sqrt{8} \cdot i
\]
\[
x - 2 = \pm 2\sqrt{2} \cdot i
\]

Finally, add 2 to both sides to solve for \(x\):
\[
x = 2 \pm 2\sqrt{2} \cdot i
\]

### Solution
The solutions are:
\[
x = 2 + 2\sqrt{2} \cdot i \quad \text{or} \quad x = 2 - 2\sqrt{2} \cdot i
\]

These solutions are complex numbers since the discriminant (inside the square root) is negative.

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How would you solve \(x^2 + 6x + 10 = 0\) using the same method?
2. What is the geometric interpretation of completing the square?
3. Can all quadratic equations be solved by completing the square?
4. How does completing the square help derive the quadratic formula?
5. How would you solve a quadratic equation with a leading coefficient other than 1?

**Tip:** Completing the square is particularly useful for solving quadratic equations that do not factorize easily or for understanding the vertex form of a parabola.

Learn how to solve the quadratic equation x^2 - 4x + 12 = 0 using the method of completing the square with a detailed step-by-step solution. This problem involves complex numbers and covers key algebraic concepts including the quadratic formula. Suitable for students in Grades 10-12.

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