$$ x^2-\left(5+2\imaginaryI\right)x\left(5+5\imaginaryI=0\right) $$

It seems there are a few issues with the equation as written. Specifically:

- You have a combination of brackets that don't seem to align properly.
- It also appears that there is an extra equal sign within the parentheses, which makes the equation unclear.

Let's try to correct it. I assume you're trying to solve a quadratic equation involving a complex number. Could the equation be something like this?

\[
x^2 - (5 + 2i)x + (5 + 5i) = 0
\]

If that is the case, we can solve this quadratic equation using the quadratic formula:

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

where $ a = 1 $, $ b = -(5 + 2i) $, and $ c = 5 + 5i $.

Let's proceed with the calculation.

### Step 1: Define the coefficients
- $ a = 1 $
- $ b = -(5 + 2i) $
- $ c = 5 + 5i $

### Step 2: Compute the discriminant
The discriminant is:

\[
\Delta = b^2 - 4ac
\]

First, let's calculate $ b^2 $:

\[
b^2 = (-(5 + 2i))^2 = (5 + 2i)^2 = 25 + 20i + 4i^2 = 25 + 20i - 4 = 21 + 20i
\]

Now compute $ 4ac $:

\[
4ac = 4(1)(5 + 5i) = 20 + 20i
\]

So the discriminant $ \Delta $ is:

\[
\Delta = (21 + 20i) - (20 + 20i) = 1
\]

### Step 3: Apply the quadratic formula
Now, use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{(5 + 2i) \pm \sqrt{1}}{2}
\]

Since $ \sqrt{1} = 1 $, we get:

\[
x = \frac{(5 + 2i) \pm 1}{2}
\]

This gives us two solutions:

\[
x_1 = \frac{(5 + 2i) + 1}{2} = \frac{6 + 2i}{2} = 3 + i
\]

\[
x_2 = \frac{(5 + 2i) - 1}{2} = \frac{4 + 2i}{2} = 2 + i
\]

### Final Answer:
The solutions to the quadratic equation are:

\[
x_1 = 3 + i \quad \text{and} \quad x_2 = 2 + i
\]

Let me know if you'd like further clarification or details on any part of this solution!

---

### Related Questions:
1. How do you solve quadratic equations with real coefficients?
2. How does the discriminant affect the nature of the roots?
3. What happens if the discriminant is negative in a quadratic equation?
4. Can you explain complex number operations in more detail?
5. How do you apply the quadratic formula to real-number quadratics?

#### Tip:
When working with complex numbers in quadratic equations, ensure that you're comfortable with basic operations like addition, multiplication, and squaring of complex numbers.

This article provides a detailed solution to the quadratic equation x² - (5 + 2i)x + (5 + 5i) = 0 involving complex numbers. Learn how to apply the quadratic formula and compute the discriminant step-by-step. Suitable for high school students and above.

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