Multiple choice

If $\alpha \,& \beta $ are roots if the equation ${x^2} + 5x - 5 = 0$, then evaluate $\dfrac{1}{{{{(\alpha + 1)}^3}}} + \dfrac{1}{{{{(\beta + 1)}^3}}}$

  1. $\dfrac {7}{27}$
  2. $\dfrac {5}{27}$
  3. $\dfrac {4}{27}$
  4. $\dfrac {1}{27}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let u = alpha + 1 and v = beta + 1. The transformed equation is u^2 + 3u - 9 = 0, so u + v = -3 and uv = -9. Hence 1/u^3 + 1/v^3 = (u^3 + v^3)/(uv)^3 = 4/27.

AI explanation

Let the new roots be 1 divided by the quantity alpha plus 1 and 1 divided by the quantity beta plus 1. Using the transformation y equals 1 divided by the quantity x plus 1, we substitute x equals 1 over y minus 1 into the original equation to get 1 over y squared minus 4 over y plus 9 equals 0. This simplifies to the new quadratic 9y squared minus 4y plus 1 equals 0. The sum of the cubes of these new roots equals the cube of their sum minus 3 times their product times their sum, which calculates to 4 over 27. The result is 4/27.

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