Multiple choice

Sum of the cubes of the roots of the quadratic equation $\displaystyle { 3x }^{ 2 }-5x+2=0$

  1. $\displaystyle -\frac { 35 }{ 27 } $
  2. $\displaystyle \frac { 7 }{ 9 } $
  3. $\displaystyle \frac { 27 }{ 35 } $
  4. $\displaystyle \frac { 35 }{ 27 } $
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D Correct answer
Explanation

For 3x^2 - 5x + 2 = 0, the sum of roots (a+b) = 5/3 and product (ab) = 2/3. The sum of cubes is (a+b)^3 - 3ab(a+b) = (5/3)^3 - 3(2/3)(5/3) = 125/27 - 10/3 = 125/27 - 90/27 = 35/27.

AI explanation

For the equation 3x^2 - 5x + 2 = 0, the sum of the roots is 5/3 and the product is 2/3. We use the identity for the sum of cubes, alpha^3 + beta^3 = (alpha + beta)^3 - 3(alpha)(beta)(alpha + beta). Substituting the values, we get (5/3)^3 - 3(2/3)(5/3), which equals 125/27 - 10/3. Converting to a common denominator gives 125/27 - 90/27, resulting in a sum of 35/27.

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