Question

Solve the equation by completing the square.
x squared minus 8 x plus 20 equals 0
Question content area bottom
Part 1
A.StartSet 4 plus or minus 4 i EndSet
StartSet 4 plus or minus 4 i EndSet
B.StartSet 6 comma 2 EndSet
StartSet 6 comma 2 EndSet
C.StartSet 4 plus or minus 2 i EndSet
StartSet 4 plus or minus 2 i EndSet
D.StartSet 4 plus 2 i EndSet
StartSet 4 plus 2 i EndSet

Answer 1

To solve the equation \( x^2 - 8x + 20 = 0 \) by completing the square, we follow these steps:

1. **Rearranging the equation**: Start with:
\[
x^2 - 8x + 20 = 0
\]
We can rewrite it as:
\[
x^2 - 8x = -20
\]

2. **Completing the square**: To complete the square for the expression \( x^2 - 8x \):
- Take half of the coefficient of \( x \) (which is -8), square it, and add it to both sides. Half of -8 is -4, and squaring it gives 16.
- Add 16 to both sides:
\[
x^2 - 8x + 16 = -20 + 16
\]
This simplifies to:
\[
x^2 - 8x + 16 = -4
\]

3. **Rewrite as a square**:
The left-hand side can be factored as:
\[
(x - 4)^2 = -4
\]

4. **Taking the square root of both sides**:
\[
x - 4 = \pm \sqrt{-4}
\]
Since \( \sqrt{-4} = 2i \):
\[
x - 4 = \pm 2i
\]

5. **Solving for \( x \)**:
Adding 4 to both sides gives:
\[
x = 4 \pm 2i
\]

Thus, the solutions can be expressed as:
\[
x = 4 + 2i \quad \text{and} \quad x = 4 - 2i
\]

The final answer is:
\[
\{ 4 + 2i, 4 - 2i \}
\]

Among the provided choices, the correct option is:
**D. \( \{ 4 + 2i \} \)**.