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The equation provided is:

\[
x^2 - 8x + 17 = 0
\]

We can solve this using the quadratic formula, which is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For this equation:

- \(a = 1\) (coefficient of \(x^2\))
- \(b = -8\) (coefficient of \(x\))
- \(c = 17\) (constant term)

Now, let's calculate the discriminant:

\[
\Delta = b^2 - 4ac = (-8)^2 - 4(1)(17) = 64 - 68 = -4
\]

Since the discriminant is negative, the solutions will be complex. Using the quadratic formula:

\[
x = \frac{-(-8) \pm \sqrt{-4}}{2(1)} = \frac{8 \pm 2i}{2}
\]

Simplifying:

\[
x = 4 \pm i
\]

So the two solutions are:

\[
x = 4 + i, \quad x = 4 - i
\]

Therefore, the solutions are:

\[
4 + i, 4 - i
\]

Would you like further details or have any questions?

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Here are five related questions:
1. How do we know when to expect complex solutions in a quadratic equation?
2. What does the discriminant tell us about the nature of the roots?
3. Can you apply the quadratic formula to solve a real-world problem?
4. How can complex numbers be represented graphically?
5. How do you simplify expressions with imaginary numbers?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) first when solving a quadratic equation. It helps determine whether the roots will be real or complex.

Use the quadratic formula to solve the equation x^2 - 8x + 17 = 0.

Learn how to solve the quadratic equation x^2 - 8x + 17 using the quadratic formula. This problem involves complex solutions and covers essential algebraic concepts, including quadratic equations and the use of the discriminant to identify complex roots. Suitable for Grades 8-10.

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