Multiple choice

If $x_1$ and $x_2$ are the real roots of the equation $x^2-kx+c=0$ then the distance between the point $A(x_1, 0)$ and $B(x_2, 0)$ is

  1. $\sqrt{k^2+4c}$
  2. $\sqrt{k^2-c}$
  3. $\sqrt{c-k^2}$
  4. $\sqrt{k^2-4c}$
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D Correct answer
Explanation

The distance between roots x1 and x2 is |x1 - x2|. Using the quadratic formula, x = (k +/- sqrt(k^2 - 4c)) / 2. The difference is sqrt(k^2 - 4c).

AI explanation

The distance between the points A(x1, 0) and B(x2, 0) is the absolute value of their difference, written as |x1 - x2|. For the quadratic equation x^2 - kx + c = 0, the sum of the roots is k and the product is c. We square the distance to get (x1 - x2)^2, which equals (x1 + x2)^2 - 4(x1)(x2) by algebraic identity. Substituting the sum and product gives k^2 - 4c, so the distance is the square root of k^2 - 4c, giving sqrt(k^2-4c).

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