The roots of the equation `(b-c) x^2 +(c-a)x+(a-b)=0`are

Options

A. 1, -1
B. 1, `((a-c))/((b-c))`
C. 1, `((a-b))/((b-c))`
D. 1, `((b-a))/((b-c))`

Answer

To find the roots of the quadratic equation \((b-c)x^2 + (c-a)x + (a-b) = 0\), we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 1: Identify coefficients
In our equation, we can identify the coefficients as follows:
- \(a = b - c\)
- \(b = c - a\)
- \(c = a - b\)

### Step 2: Substitute coefficients into the quadratic formula
Substituting the coefficients into the quadratic formula, we have:

\[ x = \frac{-(c-a) \pm \sqrt{(c-a)^2 - 4(b-c)(a-b)}}{2(b-c)} \]

### Step 3: Simplify the expression
Now, we simplify the expression under the square root:

\[ (c-a)^2 = c^2 - 2ac + a^2 \]

Next, we calculate \(4(b-c)(a-b)\):

\[ 4(b-c)(a-b) = 4((b-c)(a-b)) = 4[(b)(a) - (b)(b) - (c)(a) + (c)(b)] \]

### Step 4: Combine the terms
Now, combine the terms in the square root:

\[ x = \frac{-(c-a) \pm \sqrt{c^2 - 2ac + a^2 - 4(ab - ac - b^2 + bc)}}{2(b-c)} \]

### Step 5: Further simplification
Continuing to simplify the expression under the square root, we combine like terms:

\[ x = \frac{-(c-a) \pm \sqrt{(a^2 + c^2 - 4ab + 4bc - 2ac)}}{2(b-c)} \]

### Step 6: Finalize the roots
Now we can express the roots as:

\[ x_1 = \frac{c - a + \sqrt{(a - 2b + c)^2}}{2(b-c)} \]
\[ x_2 = \frac{c - a - \sqrt{(a - 2b + c)^2}}{2(b-c)} \]

### Step 7: Simplify the roots
This leads us to:

\[ x_1 = \frac{2a - 2b}{2(b-c)} = \frac{a - b}{b - c} \]
\[ x_2 = \frac{2b - 2c}{2(b-c)} = \frac{b - c}{b - c} = 1 \]

### Conclusion
Thus, the roots of the equation \((b-c)x^2 + (c-a)x + (a-b) = 0\) are:

\[ x_1 = \frac{a - b}{b - c}, \quad x_2 = 1 \]