Multiple choice

If the equation $x^{2}-x-12=0$ and $kx^{2}+10x+3=0$ may have one common root, then $k=$

  1. $3$ or $-\dfrac{13}{16}$
  2. $-3$ or $-\dfrac{43}{16}$
  3. $3$ or $-\dfrac{43}{16}$
  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Roots of x^2 - x - 12 = 0 are x = 4 and x = -3. If 4 is a common root: k(16) + 10(4) + 3 = 0 => 16k = -43 => k = -43/16. If -3 is a common root: k(9) + 10(-3) + 3 = 0 => 9k = 27 => k = 3.

AI explanation

Let alpha be the common root of x^2 - x - 12 = 0 and kx^2 + 10x + 3 = 0. Solving the first equation yields alpha = 4 or alpha = -3. Substituting x = 4 into the second equation gives k(16) + 40 + 3 = 0, so 16k = -43 and k = -43/16. Substituting x = -3 gives k(9) - 30 + 3 = 0, so 9k = 27 and k = 3. The possible values for k are 3 and -43/16.

Keep practicing — related questions