Multiple choice

If the roots of the equation ${ 12x }^{ 2 }+mx+5=0$ are in the ratio $5:4$ then $m=$

  1. $9\sqrt { 3 } $
  2. $\underline { + } 9\sqrt { 3 } $
  3. $6\sqrt { 3 } $
  4. $\underline { + } 6\sqrt { 3 } $
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B Correct answer
AI explanation

Let the roots be 5k and 4k. Using Vieta's formulas, the sum of the roots is 9k = -m/12 and the product is 20k^2 = 5/12. From the product equation, we find k^2 = 1/48, which means k = + or - 1/(4*sqrt(3)). Substituting this into the sum equation gives m = -12 * 9k = -108k. For k = 1/(4*sqrt(3)), m is -9*sqrt(3), and for k = -1/(4*sqrt(3)), m is 9*sqrt(3). The two possible values for m are + or - 9*sqrt(3).

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