Question

Roots of the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) are:

• A. \( \frac{a(b - c)}{c(a - b)} \)
• B. \( \frac{b(c - a)}{c(a - b)} \)
• C. \( \frac{c(a - b)}{a(b - c)} \)
• D. \( \frac{c(a - b)}{b(c - a)} \)

Hint:

Use the quadratic formula to find the roots of a quadratic equation. For simplification, calculate the discriminant first.

Answer 1

The Correct Option is C

We are given the quadratic equation: \[ a(b - c)x^2 + b(c - a)x + c(a - b) = 0. \] This is a quadratic equation of the form \( Ax^2 + Bx + C = 0 \), where: \[ A = a(b - c), \quad B = b(c - a), \quad C = c(a - b). \] The roots of a quadratic equation \( Ax^2 + Bx + C = 0 \) are given by the quadratic formula: \[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}. \] 
Step 1: First, calculate the discriminant \( \Delta \): \[ \Delta = B^2 - 4AC. \] Substitute the values of \( A \), \( B \), and \( C \): \[ \Delta = [b(c - a)]^2 - 4[a(b - c)][c(a - b)]. \] Simplifying the discriminant: \[ \Delta = b^2(c - a)^2 - 4ac(b - c)(a - b). \] This discriminant is non-negative, indicating real roots. 
Step 2: Now, using the quadratic formula, we can find the roots of the equation: \[ x = \frac{-b(c - a) \pm \sqrt{b^2(c - a)^2 - 4ac(b - c)(a - b)}}{2a(b - c)}. \] After simplifying, the roots of the quadratic equation are: \[ x = \frac{c(a - b)}{a(b - c)}. \] 
Conclusion: Thus, the roots of the equation are \( \frac{c(a - b)}{a(b - c)} \).