To solve the equation \( X + 3X - X^2 = 24 \), we first simplify it by combining like terms.

1. Combine the \( X \) terms:
   \[
   X + 3X = 4X
   \]
   So the equation becomes:
   \[
   4X - X^2 = 24
   \]

2. Rearrange the equation into standard quadratic form:
   \[
   -X^2 + 4X - 24 = 0
   \]

3. Multiply the entire equation by -1 to make the coefficient of \( X^2 \) positive:
   \[
   X^2 - 4X + 24 = 0
   \]

Next, we solve the quadratic equation \( X^2 - 4X + 24 = 0 \) using the quadratic formula \( X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -4 \), and \( c = 24 \).

4. Calculate the discriminant:
   \[
   \Delta = b^2 - 4ac = (-4)^2 - 4(1)(24) = 16 - 96 = -80
   \]

Since the discriminant is negative (\( \Delta = -80 \)), there are no real solutions. Instead, we have complex solutions.

5. Find the complex solutions using the quadratic formula:
   \[
   X = \frac{-(-4) \pm \sqrt{-80}}{2(1)} = \frac{4 \pm \sqrt{-80}}{2}
   \]

   \[
   X = \frac{4 \pm \sqrt{80}i}{2} = \frac{4 \pm 4\sqrt{5}i}{2} = 2 \pm 2\sqrt{5}i
   \]

Therefore, the solutions to the equation \( X + 3X - X^2 = 24 \) are:
\[
X = 2 + 2\sqrt{5}i \quad \text{and} \quad X = 2 - 2\sqrt{5}i
\]

Would you like more details or have any questions?

Here are some additional questions you might want to ask:
1. How do you derive the quadratic formula?
2. What are the steps to complete the square?
3. How do you solve a quadratic equation with complex solutions?
4. Can you provide examples of other quadratic equations?
5. What is the importance of the discriminant in solving quadratic equations?

**Tip:** When solving quadratic equations, always check the discriminant first to determine the nature of the roots.

Learn how to solve the equation X + 3X - X^2 = 24 with complex solutions. This problem involves using the quadratic formula and understanding complex numbers. Suitable for advanced high school students in Grades 11-12.

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