Multiple choice

If $\alpha,\beta$ are the roots of the quadratic equation $4x^{2}-4x+1=0$, then $\alpha^{3}+\beta^{3}$ is equal to

  1. $1/4$
  2. $1/8$
  3. $16$
  4. $32$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Roots of 4x^2 - 4x + 1 = 0 are (2x-1)^2 = 0, so x = 1/2. Thus alpha = 1/2, beta = 1/2. alpha^3 + beta^3 = 1/8 + 1/8 = 2/8 = 1/4.

AI explanation

Using the identity for the sum of roots, alpha plus beta equals negative b divided by a, giving 4 divided by 4, which is 1. The product of the roots, alpha times beta, equals c divided by a, giving 1 divided by 4. Apply the algebraic identity for the sum of cubes, alpha cubed plus beta cubed equals (alpha plus beta) times (alpha squared minus alpha times beta plus beta squared), which simplifies to (alpha plus beta) times ((alpha plus beta) squared minus 3 times alpha times beta). Substitute the known values to get 1 times (1 squared minus 3 divided by 4), resulting in 1 divided by 4.

Keep practicing — related questions