## Solving the Quadratic Equation: (a² + b²)x² - 2(ac + bd)x + (c² + d²) = 0

This equation, (a² + b²)x² - 2(ac + bd)x + (c² + d²) = 0, is a quadratic equation in the form of ax² + bx + c = 0. Here's how to solve it and understand its significance.

### Understanding the Equation

**Coefficients:**The coefficients in this equation are:**a = a² + b²****b = -2(ac + bd)****c = c² + d²**

**Significance:**This equation has a special structure that arises in various mathematical and physical contexts.

### Solving the Equation

We can solve for 'x' using the quadratic formula:

**x = [-b ± √(b² - 4ac)] / 2a**

Substituting the coefficients from our equation:

**x = [2(ac + bd) ± √((-2(ac + bd))² - 4(a² + b²)(c² + d²))] / 2(a² + b²)**

### Simplifying the Solution

The expression under the square root can be simplified:

**(-2(ac + bd))² = 4(a²c² + 2abcd + b²d²)****4(a² + b²)(c² + d²) = 4(a²c² + a²d² + b²c² + b²d²)**

Therefore, the discriminant (b² - 4ac) becomes:

**4(a²c² + 2abcd + b²d²) - 4(a²c² + a²d² + b²c² + b²d²) = -4(a²d² - 2abcd + b²c²) = -4(ad - bc)²**

The final solution is:

**x = [(ac + bd) ± √(-(ad - bc)²)] / (a² + b²)**

**x = [(ac + bd) ± i(ad - bc)] / (a² + b²) (where 'i' is the imaginary unit)**

### Key Points

**Real Solutions:**If (ad - bc)² is positive, the equation has two distinct real solutions.**Complex Solutions:**If (ad - bc)² is negative, the equation has two complex conjugate solutions.**Special Case:**If (ad - bc) = 0, the equation has a single repeated real solution.

### Applications

This quadratic equation finds applications in various fields, including:

**Geometry:**In problems related to circles, ellipses, and other conic sections.**Physics:**In describing the motion of objects under constant acceleration.**Linear Algebra:**In solving systems of linear equations.

### Conclusion

Understanding and solving quadratic equations like (a² + b²)x² - 2(ac + bd)x + (c² + d²) = 0 is crucial for various mathematical and scientific disciplines. The unique structure of this equation reveals interesting insights into the nature of solutions and its applications in real-world problems.