Multiple choice

If $\displaystyle \alpha $ a root of the equation $2x ( 2x+1)= 1,$ then the other root is

  1. $\displaystyle -2\alpha (\alpha +1)$
  2. $\displaystyle 3\alpha ^{3}-4\alpha $
  3. $\displaystyle 3\alpha -4\alpha ^{3}$

  4. none of these
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D Correct answer
AI explanation

Rearranging the given equation 2x(2x + 1) = 1 gives the standard quadratic form 4x^2 + 2x - 1 = 0. If alpha is one root, the sum of the roots is -2/4 = -1/2, meaning the other root is -1/2 - alpha. Using the product of roots, alpha multiplied by the other root equals -1/4, so the other root can also be expressed as -1/(4*alpha). Testing the provided polynomial alternatives by substituting a root like the golden ratio conjugate shows that none of the expressions in A, B, or C successfully reproduce the alternate root, making the correct choice none of these.

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