Which one of the following is a root of the equation?

Direction: Consider the following for the two (02) items that follow :<br>The roots of the quadratic equation a^{2}(b^{2}-c^{2})x^{2}+b^{2}(c^{2}-a^{2})x+c^{2}(a^{2}-b^{2})=0 are equal (a^{2} \neq b^{2} \neq c^{2})

  1. A. \frac{b^{2}(c^{2}-a^{2})}{a^{2}(c^{2}-b^{2})}
  2. B. \frac{b^{2}(c^{2}-a^{2})}{a^{2}(b^{2}-c^{2})}
  3. C. \frac{b^{2}(c^{2}-a^{2})}{2a^{2}(c^{2}-b^{2})} ✓
  4. D. \frac{b^{2}(c^{2}-a^{2})}{2a^{2}(b^{2}-c^{2})}

Correct Answer: C. \frac{b^{2}(c^{2}-a^{2})}{2a^{2}(c^{2}-b^{2})}

Explanation

As established, the roots are equal to 1. The sum of the roots is -\frac{b^2(c^2-a^2)}{a^2(b^2-c^2)}. Since 1+1=2, we have -\frac{b^2(c^2-a^2)}{a^2(b^2-c^2)} = 2. Therefore, 1 = \frac{-b^2(c^2-a^2)}{2a^2(b^2-c^2)} = \frac{b^2(c^2-a^2)}{2a^2(c^2-b^2)}.

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