To solve the quadratic equation:

$x^2 - 6x + 25 = 0$

We can use the quadratic formula:

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

In this case, $a = 1$, $b = -6$, and $c = 25$. Let's substitute these values into the quadratic formula:

$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(25)}}{2(1)}$

Simplify step by step:

$x = \frac{6 \pm \sqrt{36 - 100}}{2}$

$x = \frac{6 \pm \sqrt{-64}}{2}$

$x = \frac{6 \pm 8i}{2}$

$x = 3 \pm 4i$

So, the solutions are:

$x = 3 + 4i \quad \text{and} \quad x = 3 - 4i$

These are complex solutions.

Would you like more details on any part of this, or have any further questions?

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Here are 5 related questions to deepen your understanding:

1. How do you graph a quadratic function with complex roots?
2. What is the significance of the discriminant in determining the nature of roots?
3. Can you apply the quadratic formula to any quadratic equation?
4. How do you simplify expressions involving complex numbers?
5. What are the geometric interpretations of complex numbers?

Tip: When the discriminant ($b^2 - 4ac$) is negative, the quadratic equation will have complex roots.