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The diagram you've uploaded shows a sector of a circle with center \( S \), where \( \angle PST \) is a central angle, and \( \angle PSA \) and \( \angle AST \) are inscribed angles subtended by arcs \( \overarc{PT} \) and \( \overarc{PS} \), respectively. The angles provided are:

- Central angle \( \angle PST = (24x + 50)^\circ \)
- Inscribed angle \( \angle PSA = (x^2)^\circ \)

### Key Properties:
1. **Central Angle Theorem:** The central angle is twice any inscribed angle that subtends the same arc.
   \[
   \text{Thus, } (24x + 50)^\circ = 2 \times (x^2)^\circ
   \]
2. **Set up the equation:**
   \[
   24x + 50 = 2x^2
   \]

### Solve the quadratic equation:
\[
2x^2 - 24x - 50 = 0
\]

### Simplify by dividing by 2:
\[
x^2 - 12x - 25 = 0
\]

### Solve using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where \( a = 1 \), \( b = -12 \), and \( c = -25 \).

\[
x = \frac{12 \pm \sqrt{(-12)^2 - 4(1)(-25)}}{2(1)}
\]
\[
x = \frac{12 \pm \sqrt{144 + 100}}{2}
\]
\[
x = \frac{12 \pm \sqrt{244}}{2}
\]
\[
x = \frac{12 \pm \sqrt{4 \times 61}}{2}
\]
\[
x = \frac{12 \pm 2\sqrt{61}}{2}
\]
\[
x = 6 \pm \sqrt{61}
\]

### Possible values for \( x \):
\[
x = 6 + \sqrt{61} \quad \text{or} \quad x = 6 - \sqrt{61}
\]

Since \( x \) represents an angle measurement in degrees, and should be a positive value, we typically consider the positive root \( x = 6 + \sqrt{61} \) as the solution. 

However, depending on the context (like reasonable angle range), the other root might also be valid. But in practical scenarios, we usually take the value that results in a positive angle.

Do you want further details or clarifications on this solution?

### Related Questions:
1. How do you derive the quadratic formula?
2. What are the properties of central and inscribed angles in a circle?
3. How can you factorize a quadratic equation like this one?
4. What are other methods for solving quadratic equations?
5. How does the discriminant determine the nature of the roots in a quadratic equation?

### Tip:
Always check whether the solutions to your quadratic equation make sense in the context of the problem, especially in geometric contexts where the solution often must be a positive number or within a certain range.

Learn how to solve for x in a geometry problem involving central and inscribed angles in a circle. Includes detailed steps and explanations using the central angle theorem and quadratic formula. Suitable for high school students in Grades 10-12.

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