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.