Let's solve the equation step by step.

Given Equation:

$\text{sup}\left((2x - 5)^2\right) + 75 = -275$

Step 1: Isolate the squared term

Subtract 75 from both sides: $(2x - 5)^2 = -275 - 75$ $(2x - 5)^2 = -350$

Step 2: Take the square root of both sides

Since the right side is negative, the solutions will be complex numbers: $2x - 5 = \pm \sqrt{-350}$ $2x - 5 = \pm \sqrt{350} \cdot i$

Step 3: Simplify the square root

The square root of 350 can be simplified: $\sqrt{350} = \sqrt{25 \times 14} = 5\sqrt{14}$ So, the equation becomes: $2x - 5 = \pm 5\sqrt{14} \cdot i$

Step 4: Solve for $x$

Add 5 to both sides: $2x = 5 \pm 5\sqrt{14} \cdot i$ Finally, divide by 2: $x = \frac{5 \pm 5\sqrt{14} \cdot i}{2}$

Step 5: Write the solutions

The two solutions are: $x_1 = \frac{5 + 5\sqrt{14} \cdot i}{2}$ $x_2 = \frac{5 - 5\sqrt{14} \cdot i}{2}$

So, the solutions are:

$x_1 = \frac{5}{2} + \frac{5\sqrt{14}}{2}i$ $x_2 = \frac{5}{2} - \frac{5\sqrt{14}}{2}i$

If you have any questions or want further details, feel free to ask!

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Related Questions:

1. What are the square root properties used in complex numbers?
2. How do you simplify square roots involving prime factors?
3. What are the properties of complex numbers and their standard forms?
4. How can we verify complex number solutions for quadratic equations?
5. What are the methods for solving quadratic equations with real coefficients?

Tip: Always check if the equation can result in complex solutions by considering the sign of the squared term.