Multiple choice

If $\displaystyle \alpha ,\beta $ are the roots of the equation $\displaystyle { x }^{ 2 }-x-4=0$, find the value of $\displaystyle \frac { 1 }{ \alpha } +\frac { 1 }{ \beta } -\alpha \beta $.

  1. $\displaystyle \frac { 4 }{ 15 } $
  2. $\displaystyle \frac { 15 }{ 4 } $
  3. $\displaystyle \frac { 4 }{ 5 } $
  4. $\displaystyle \frac { 5 }{ 4 } $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For x^2 - x - 4 = 0, sum of roots (alpha + beta) = 1, product (alpha * beta) = -4. The expression is (alpha + beta) / (alpha * beta) - (alpha * beta) = 1 / -4 - (-4) = -0.25 + 4 = 3.75 = 15/4.

AI explanation

According to Vieta's formulas for the equation x^2 - x - 4 = 0, the sum of the roots alpha + beta equals 1 and the product alpha * beta equals -4. The expression 1/alpha + 1/beta can be combined into a single fraction as (alpha + beta) / (alpha * beta), which evaluates to 1 / (-4) = -1/4. The expression also includes the term -alpha * beta, which evaluates to -(-4) = 4. Adding these two parts together gives -1/4 + 4 = -1/4 + 16/4 = 15/4.

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