The equation `(b-c) x^(2)+ (c-a) x+(a-b)=0` has

Options

A. equal roots
B. irrational roots
C. rational roots
D. none of these

Answer

To solve the equation \((b-c)x^2 + (c-a)x + (a-b) = 0\) and determine the nature of its roots, we will use the discriminant method. The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula:

\[ D = b^2 - 4ac \]

### Step 1: Identify coefficients
In our equation, we have:
- \(a = b - c\)
- \(b = c - a\)
- \(c = a - b\)

### Step 2: Calculate the discriminant
Now we will substitute these values into the discriminant formula:

\[ D = (c - a)^2 - 4(b - c)(a - b) \]

### Step 3: Expand the discriminant
We will expand both parts of the discriminant:

1. **Calculate \((c - a)^2\):**
\[ (c - a)^2 = c^2 - 2ac + a^2 \]

2. **Calculate \(4(b - c)(a - b)\):**
\[ 4(b - c)(a - b) = 4[(b - c)(a - b)] = 4[ba - b^2 - ac + bc] \]

### Step 4: Combine the terms
Now we can substitute these expansions back into the discriminant:

\[ D = (c^2 - 2ac + a^2) - 4(ba - b^2 - ac + bc) \]

### Step 5: Simplify the discriminant
Combine like terms:

\[ D = c^2 - 2ac + a^2 - 4ba + 4b^2 + 4ac - 4bc \]

Rearranging gives:

\[ D = a^2 + c^2 + 4b^2 + 2ac - 4ab - 4bc \]

### Step 6: Analyze the discriminant
To determine the nature of the roots, we need to check the discriminant:

- If \(D > 0\), the roots are real and distinct.
- If \(D = 0\), the roots are real and equal.
- If \(D < 0\), the roots are complex.

### Step 7: Check if \(D\) is a perfect square
From the analysis, we can see that \(D\) can be expressed as a perfect square:

\[ D = (a + 2b + c)^2 \]

Since \(D\) is a perfect square, it implies that \(D \geq 0\).

### Conclusion
Since \(D\) is a perfect square, the roots of the quadratic equation \((b-c)x^2 + (c-a)x + (a-b) = 0\) are rational.

### Final Answer
The equation has rational roots.

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