Multiple choice

If α and β are the roots of the equation 2x(2x + 1) = 1, then β is equal to:

  1. 2α2

  2. -2α(α + 1)

  3. 2α(α - 1)

  4. 2α(α + 1)
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B Correct answer
Explanation

The equation is 4x^2 + 2x - 1 = 0. The sum of roots α + β = -2/4 = -1/2, so β = -1/2 - α. The product of roots αβ = -1/4, so β = -1/(4α). Testing the options, if β = -2α(α + 1), then β = -2α^2 - 2α. Since 4α^2 + 2α - 1 = 0, we have 2α^2 = 0.5 - α. Substituting this: β = -(0.5 - α) - 2α = -0.5 - α, which matches the sum of roots condition.

AI explanation

Expanding and rearranging the equation gives 4x^2 + 2x - 1 = 0. Since a is a root, we can substitute x = a to get 4a^2 = -2a + 1, which means 4a^2 + 2a = 1. By Vieta's formulas for the roots a and b, the sum is a + b = -1/2, so b = -1/2 - a. Substituting -2a for b using the previous relation yields b = -2a(a + 1).

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