Multiple choice

On solving the following quadratic equation by factorization, the roots are $\dfrac{\sqrt3}{4}, \, -\dfrac{2}{\sqrt3}$ $4\sqrt{3}x^2 \, + \, 5x \, - \, 2\sqrt{3} \, = \, 0$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation is 4sqrt(3)x^2 + 5x - 2sqrt(3) = 0. Roots are given by quadratic formula: x = [-5 +/- sqrt(25 - 4(4sqrt(3))(-2sqrt(3)))] / 8sqrt(3) = [-5 +/- sqrt(25 + 96)] / 8sqrt(3) = [-5 +/- 11] / 8sqrt(3). Roots are 6 / 8sqrt(3) = 3 / 4sqrt(3) = sqrt(3)/4 and -16 / 8sqrt(3) = -2/sqrt(3).

AI explanation

For the equation 4*sqrt(3)x^2 + 5x - 2*sqrt(3) = 0, split the middle term to get 4*sqrt(3)x^2 + 8x - 3x - 2*sqrt(3) = 0. Factoring by grouping gives 4x(sqrt(3)x + 2) - sqrt(3)(sqrt(3)x + 2) = 0, leading to factors (4x - sqrt(3))(sqrt(3)x + 2) = 0. Setting each factor to zero yields the roots x = sqrt(3)/4 and x = -2/sqrt(3). The statement is True.

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