Multiple choice

If the equations $ x^{2}-11x+a=0 $ and $ x^{2}-14x+2a=0 $ have a common root, then $a =$

  1. $0, 18$
  2. $0, 30$
  3. $18, 30$
  4. $0, 24$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If they have a common root r, then r^2 - 11r + a = 0 and r^2 - 14r + 2a = 0. Subtracting the equations: 3r - a = 0, so a = 3r. Substitute a = 3r into the first: r^2 - 11r + 3r = 0 => r^2 - 8r = 0. So r = 0 or r = 8. If r = 0, a = 0. If r = 8, a = 24.

AI explanation

Let the common root be p, so p squared minus 11p plus a equals 0 and p squared minus 14p plus 2a equals 0. Subtracting the second equation from the first gives 3p minus a equals 0, meaning a equals 3p. Substituting p equals a divided by 3 into the first equation yields a squared divided by 9 minus 11a divided by 3 plus a equals 0, which factors to a times (a minus 24) equals 0. The result is 0 and 24.

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