Multiple choice

If $a$ and $b$ are the roots of the quadratic equation $\displaystyle { 3x }^{ 2 }+8x+2=0$, find the value of $\displaystyle { a }^{ 3 }+{ b }^{ 3 }$.

  1. $\dfrac{368}{27}$
  2. $\dfrac{-368}{27}$
  3. $\dfrac{-368}{24}$

  4. None of these
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B Correct answer
Explanation

For 3x^2 + 8x + 2 = 0, a + b = -8/3 and ab = 2/3. a^3 + b^3 = (a+b)^3 - 3ab(a+b) = (-8/3)^3 - 3(2/3)(-8/3) = -512/27 + 16/3 = -512/27 + 144/27 = -368/27.

AI explanation

From the given equation 3x^2+8x+2=0, the sum of roots a+b is -8/3 and the product ab is 2/3. Using the algebraic identity a^3+b^3 = (a+b)^3-3ab(a+b), we substitute the values to get (-8/3)^3-3*(2/3)*(-8/3). This simplifies to -512/27 + 48/9, which equals -512/27 + 144/27, giving the final result of -368/27.

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