Multiple choice

Solving the following quadratic equations by factorization method: $ 6x^{2}-17ix-12 = 0 $

  1. $ \dfrac{3}{2}i,\dfrac{4}{3}i $
  2. $ 2i, 3i$
  3. $ -\dfrac{3}{2}i,\dfrac{4}{3}i $
  4. $ 3i, 4i $
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A Correct answer
Explanation

The equation is 6x^2 - 17ix - 12 = 0. Using the quadratic formula: x = [17i +/- sqrt((-17i)^2 - 4(6)(-12))] / 12 = [17i +/- sqrt(-289 + 288)] / 12 = [17i +/- i] / 12. Roots are 18i/12 = 3i/2 and 16i/12 = 4i/3.

AI explanation

To factorize 6x^2 - 17ix - 12 = 0 by splitting the middle term, we find two numbers whose product is (6)(-12) = -72 and whose sum is -17i. These numbers are -9i and 8i, so we rewrite the middle term to get 6x^2 - 9ix + 8ix - 12 = 0. Factoring by grouping gives 3ix(2x - 3i) + 4(2x - 3i) = 0, which results in (3ix + 4)(2x - 3i) = 0. Equating each factor to zero, x = -4/(3i) and x = 3i/2, which are equivalent to (4/3)i and (3/2)i.

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