Multiple choice

The values of he equation $a$for which both the roots of the equation $(a-6)x^2=a(x-3)$ are positive are given by

  1. $(0,6)$
  2. $\displaystyle \left( 0,\frac { 72 }{ 11 }  \right) $
  3. $\displaystyle \left( 6,\frac { 72 }{ 11 }  \right) $
  4. $\displaystyle \left( \frac { 72 }{ 13 } ,6 \right) $
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C Correct answer
AI explanation

Rearranging the equation (a-6)x^2 = a(x-3) gives the standard quadratic form (a-6)x^2 - ax + 3a = 0. For both roots to be positive, the product of the roots must be positive, so 3a / (a-6) > 0, which yields a > 6. The sum of the roots must also be positive, so a / (a-6) > 0, which is satisfied for a > 6. Finally, for real roots, the discriminant must be positive, meaning (-a)^2 - 4(a-6)(3a) = a^2 - 12a^2 + 72a = -11a^2 + 72a > 0, which simplifies to a < 72/11. Combining the inequalities gives the range (6, 72/11).

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