Multiple choice

The common root of the equations $x^2-bx+c=0$ and $x^2+bx-a=0$ is

  1. $\dfrac{c+a}{2b}$
  2. $\dfrac{c-a}{2b}$
  3. $\dfrac{c+b}{2a}$
  4. $\dfrac{a+b}{2c}$
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A Correct answer
Explanation

Let x be the common root. x^2 - bx + c = 0 and x^2 + bx - a = 0. Subtracting: -2bx + c + a = 0. 2bx = c + a. x = (c + a) / 2b.

AI explanation

Assume r is the common root for x^2-bx+c=0 and x^2+bx-a=0. This gives r^2-br+c=0 and r^2+br-a=0. Subtracting the second equation from the first results in -2br+c+a=0. Isolating r gives 2br=c+a, which simplifies to r=(c+a)/(2b).

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