Multiple choice

In the following, determine whether the given quadratic equation have real roots and if so, find the roots : $\sqrt{3}x^2 \, + \, 10x \, - \, 8\sqrt{3} \, = \, 0$

  1. $-4\sqrt3, \, \dfrac{2}{\sqrt 3}$
  2. $4\sqrt3, \, \dfrac{2}{\sqrt 3}$
  3. $-4\sqrt3, \, \dfrac{-2}{\sqrt 3}$

  4. not real
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / 2a. Here a = sqrt(3), b = 10, c = -8sqrt(3). Discriminant = 100 - 4(sqrt(3))(-8sqrt(3)) = 100 + 96 = 196. x = (-10 +/- 14) / 2sqrt(3). Roots are 4/2sqrt(3) = 2/sqrt(3) and -24/2sqrt(3) = -12/sqrt(3) = -4sqrt(3).

AI explanation

Using the quadratic formula for sqrt(3)x^2 + 10x - 8(sqrt(3)) = 0, the discriminant is 10^2 - 4(sqrt(3))(-8(sqrt(3))) = 100 + 96 = 196. The roots are calculated as (-10 ± sqrt(196)) / (2(sqrt(3))) = (-10 ± 14) / (2(sqrt(3))). This gives the two roots as -12 / (2(sqrt(3))) which simplifies to -4(sqrt(3)), and 4 / (2(sqrt(3))) which simplifies to 2 / sqrt(3).

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