To solve the equation \( x^2 + 5 = x - 12 \), follow these steps:

1. **Move all terms to one side of the equation** to set it equal to zero:
   \[
   x^2 - x + 5 + 12 = 0
   \]
   Simplify this to:
   \[
   x^2 - x + 17 = 0
   \]

2. **Solve the quadratic equation** using the quadratic formula. The quadratic formula is given by:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   For the equation \( x^2 - x + 17 = 0 \), \( a = 1 \), \( b = -1 \), and \( c = 17 \).

3. **Calculate the discriminant** \( \Delta = b^2 - 4ac \):
   \[
   \Delta = (-1)^2 - 4(1)(17) = 1 - 68 = -67
   \]
   Since the discriminant is negative, the equation has **no real solutions**. However, it has two complex solutions.

4. **Find the complex solutions**:
   \[
   x = \frac{-(-1) \pm \sqrt{-67}}{2(1)} = \frac{1 \pm \sqrt{67}i}{2}
   \]

Thus, the solutions are:
\[
x = \frac{1 + \sqrt{67}i}{2}, \quad x = \frac{1 - \sqrt{67}i}{2}
\]

Would you like more details, or do you have any questions?

Here are 5 related questions you might find interesting:
1. How do you determine the nature of the roots of a quadratic equation?
2. Can you explain why a negative discriminant results in complex solutions?
3. What is the geometric interpretation of complex roots?
4. How would you solve the quadratic equation if the discriminant were zero?
5. Can you solve a quadratic equation using methods other than the quadratic formula?

**Tip:** When the discriminant is negative, the quadratic equation will always have two complex conjugate solutions.

Learn how to solve the quadratic equation x^2 + 5 = x - 12 with a detailed step-by-step solution. This problem involves complex numbers, the quadratic formula, and discusses the nature of roots in quadratic equations. Suitable for students in Grades 10-12.

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