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

This article will guide you through solving the quadratic equation: **(c²-ab)x² - 2(a²-bc)x + (b²-ac) = 0**. We'll explore the steps involved in finding the solutions for 'x' using the quadratic formula and discuss some key observations about this specific equation.

### Understanding the Quadratic Formula

The quadratic formula is a powerful tool used to solve equations of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are coefficients. The formula states:

**x = [-b ± √(b² - 4ac)] / 2a**

### Applying the Formula to Our Equation

Let's break down our equation and identify the coefficients:

- a = (c²-ab)
- b = -2(a²-bc)
- c = (b²-ac)

Now, we can substitute these values into the quadratic formula:

**x = [2(a²-bc) ± √((-2(a²-bc))² - 4(c²-ab)(b²-ac))] / 2(c²-ab)**

### Simplifying the Expression

To make the solution more manageable, we need to simplify the expression under the square root:

**x = [2(a²-bc) ± √(4(a²-bc)² - 4(c²-ab)(b²-ac))] / 2(c²-ab)**

**x = [2(a²-bc) ± 2√((a²-bc)² - (c²-ab)(b²-ac))] / 2(c²-ab)**

**x = [(a²-bc) ± √((a²-bc)² - (c²-ab)(b²-ac))] / (c²-ab)**

### Analyzing the Solution

The solution to this quadratic equation depends on the value of the expression inside the square root. Let's examine this expression further:

**((a²-bc)² - (c²-ab)(b²-ac))**

Expanding this, we get:

**a⁴ - 2a²bc + b²c² - (b²c² - abc³ - ab³c + a²c⁴)**

**a⁴ - 2a²bc + abc³ + ab³c - a²c⁴**

**a²(a² - 2bc + bc³ + b³c - c⁴)**

This expression reveals some interesting relationships between the coefficients 'a', 'b', and 'c'. The nature of the solutions (real or complex) depends on the sign of this expression.

**If the expression is positive**, the quadratic equation will have **two distinct real roots**.

**If the expression is negative**, the quadratic equation will have **two complex roots** (conjugate pairs).

**If the expression is zero**, the quadratic equation will have **one real root (a double root)**.

### Conclusion

Solving the quadratic equation (c²-ab)x² - 2(a²-bc)x + (b²-ac) = 0 involves applying the quadratic formula and carefully analyzing the resulting expression. Understanding the relationship between the coefficients 'a', 'b', and 'c' within the solution helps determine the nature of the roots (real or complex). Remember to simplify the solution wherever possible to obtain a more manageable expression.