Multiple choice

If $a,b$ are roots of the equation $x^2+px-q=0$ and $c,d$ are roots of $x^2+px+r=0$, then the value of $(a-c)(a-d)$ is-

  1. $q-r$
  2. $q+r$
  3. $p+r$
  4. $p-r$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Since a, b are roots of x^2 + px - q = 0, a^2 + pa - q = 0, so a^2 + pa = q. Since c, d are roots of x^2 + px + r = 0, c+d = -p and cd = r. (a-c)(a-d) = a^2 - a(c+d) + cd = a^2 - a(-p) + r = a^2 + pa + r. Since a^2 + pa = q, this equals q + r.

AI explanation

Since c and d are roots of x^2+px+r=0, the expression (a-c)(a-d) is identical in form to evaluating this quadratic at x=a. Substituting x=a gives a^2+pa+r. Because a is a root of x^2+px-q=0, we know a^2+pa=q. Substituting a^2+pa=q into the previous result yields q+r. The value is q+r.

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