## Analyzing the Quadratic Equation: (a-12)x² + 2(a-12)x + 2 = 0

This article explores the quadratic equation **(a-12)x² + 2(a-12)x + 2 = 0**, focusing on its properties and how to solve it.

### Understanding the Equation

The given equation is a quadratic equation in the form **ax² + bx + c = 0**, where:

**a = (a-12)****b = 2(a-12)****c = 2**

This equation has various interesting properties depending on the value of 'a'.

### Solving the Equation

We can solve this quadratic equation using the quadratic formula:

**x = (-b ± √(b² - 4ac)) / 2a**

Substituting the values from our equation:

**x = (-2(a-12) ± √( (2(a-12))² - 4(a-12)(2) ) ) / 2(a-12)**

Simplifying the equation:

**x = (-2(a-12) ± √(4(a-12)² - 8(a-12) ) ) / 2(a-12)**

**x = (-2(a-12) ± √(4(a-12)(a-12 - 2) ) ) / 2(a-12)**

**x = (-2(a-12) ± 2√((a-12)(a-14) ) ) / 2(a-12)**

**x = (- (a-12) ± √((a-12)(a-14) ) ) / (a-12)**

This gives us two possible solutions for x:

**x = (- (a-12) + √((a-12)(a-14) ) ) / (a-12)****x = (- (a-12) - √((a-12)(a-14) ) ) / (a-12)**

### Analyzing the Solutions

The solutions of this quadratic equation depend on the value of 'a'. Here's a breakdown:

**Case 1: a = 12**- If a = 12, the equation becomes
**0x² + 0x + 2 = 0**. This equation has no solutions.

- If a = 12, the equation becomes
**Case 2: a ≠ 12**- If a ≠ 12, the solutions will be real numbers as long as the discriminant (b² - 4ac) is greater than or equal to 0.
**Discriminant:**4(a-12)(a-14) ≥ 0- This inequality holds true when:
**a ≤ 12 or a ≥ 14**

This means the equation has **two distinct real solutions** if a ≤ 12 or a ≥ 14. If a is between 12 and 14, the equation has **no real solutions** but two complex solutions.

### Conclusion

The quadratic equation (a-12)x² + 2(a-12)x + 2 = 0 has solutions that depend heavily on the value of 'a'. Understanding the discriminant and its relationship to 'a' allows us to predict the nature of the solutions, be they real or complex. This analysis provides a comprehensive understanding of this equation and its various possible solutions.