Solve the following for x using the square root property.  

sup(bracket(2x-5),2)+75=-275

Enter one value per answer box. If there are more answer boxes than solutions, leave the extra boxes blank. Make sure to fully simplify any fractional coefficients, and enter complex numbers in standard form.

x=	



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.

Solve the following for x using the square root property.

sup(bracket(2x-5),2)+75=-275

Enter one value per answer box. If there are more answer boxes than solutions, leave the extra boxes blank. Make sure to fully simplify any fractional coefficients, and enter complex numbers in standard form.

x=

Master the process of solving for x using the square root property with complex number solutions. This detailed guide breaks down each step, including the simplification of radicals and the application of complex number properties. Perfect for high school students in Grades 9-12.

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