(a^2+b^2)x^2-2(ac+bd)x+(c2+d2)=0

3 min read Jun 16, 2024

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.