Multiple choice

If the equation $\displaystyle k \left ( 6x^{2} + 3 \right ) + rx + \left ( 2x^{2} - 1 \right ) = 0$ and $\displaystyle 6k\left ( 2x^{2} + 1 \right ) + px + 4x^{2} - 2 = 0$ have symmetrical roots, then $\displaystyle 2r - p$ equals

  1. $\displaystyle 1$
  2. $\displaystyle 2$
  3. $\displaystyle 3$
  4. $\displaystyle 0$
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D Correct answer
Explanation

Symmetrical roots imply the equations are proportional. Comparing coefficients of the two equations: k(6x^2 + 3) + rx + (2x^2 - 1) = 0 and 6k(2x^2 + 1) + px + 4x^2 - 2 = 0. Simplifying and equating ratios leads to 2r - p = 0.

AI explanation

Simplifying the first equation yields (6k + 2)x^2 + rx + (3k - 1) = 0 and simplifying the second yields (12k + 4)x^2 + px + (6k - 2) = 0. Notice that all coefficients of the second equation are exactly twice the corresponding coefficients of the first equation. For the two equations to have symmetrical roots, meaning their roots are equal in magnitude but opposite in sign, the sum of their roots must be zero. This requires -r / (6k + 2) = -p / (12k + 4), which simplifies to 2r = p, and therefore 2r - p equals 0.

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