Solve by completing the square.
4x Superscript 2minus16xplus17equals0
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Part 1
The solution set is StartSet nothing EndSet
  
.

Let's solve the quadratic equation by completing the square.

The equation is:

\[
4x^2 - 16x + 17 = 0
\]

### Step 1: Divide by 4
First, divide the entire equation by 4 to simplify it.

\[
x^2 - 4x + \frac{17}{4} = 0
\]

### Step 2: Move the constant term to the right-hand side
Next, move the constant term \(\frac{17}{4}\) to the right side of the equation.

\[
x^2 - 4x = -\frac{17}{4}
\]

### Step 3: Complete the square
To complete the square, we take half of the coefficient of \(x\) (which is \(-4\)), square it, and add it to both sides of the equation. Half of \(-4\) is \(-2\), and squaring it gives \(4\).

Now, add \(4\) to both sides of the equation:

\[
x^2 - 4x + 4 = -\frac{17}{4} + 4
\]

Simplify the right-hand side:

\[
x^2 - 4x + 4 = -\frac{17}{4} + \frac{16}{4}
\]

\[
x^2 - 4x + 4 = -\frac{1}{4}
\]

### Step 4: Express as a perfect square
The left-hand side is now a perfect square:

\[
(x - 2)^2 = -\frac{1}{4}
\]

### Step 5: Solve for \(x\)
Take the square root of both sides:

\[
x - 2 = \pm \sqrt{-\frac{1}{4}}
\]

Since we are taking the square root of a negative number, the solutions are complex:

\[
x - 2 = \pm \frac{i}{2}
\]

### Step 6: Solve for \(x\)

\[
x = 2 \pm \frac{i}{2}
\]

Thus, the solution set is:

\[
x = 2 + \frac{i}{2}, \quad x = 2 - \frac{i}{2}
\]

### Final answer:
The solution set is:

\[
\{ 2 + \frac{i}{2}, 2 - \frac{i}{2} \}
\]

Would you like more details or have any questions?

### Related Questions:
1. How does completing the square work in real-number solutions?
2. How can you use the quadratic formula instead of completing the square?
3. What happens when you have a different leading coefficient, like 9 instead of 4?
4. Can you explain how to graph the complex solutions?
5. How is completing the square related to solving quadratic inequalities?

### Tip:
When completing the square, always remember to take half the coefficient of \(x\), square it, and then balance the equation accordingly.

Solve by completing the square. 4x^2 - 16x + 17 = 0

Learn how to solve the quadratic equation 4x^2 - 16x + 17 = 0 by completing the square. This detailed step-by-step solution explains how to work with complex numbers, the quadratic equation, and completing the square. Suitable for students in Grades 10-12.

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