Multiple choice

Let $\alpha ,\beta $ be the roots of ${ x }^{ 2 }+3x+5=0$, then the equation whose roots are $-\cfrac { 1 }{ \alpha } $ and $-\cfrac { 1 }{ \beta } $ is

  1. $5{ x }^{ 2 }+3x-4=0$
  2. $5{ x }^{ 2 }-3x+4=0$
  3. $5{ x }^{ 2 }+3x-1=0$
  4. $5{ x }^{ 2 }-3x+1=0$
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D Correct answer
Explanation

If roots are -1/alpha and -1/beta, we substitute x = -1/y into the original equation x^2 + 3x + 5 = 0. This gives (-1/y)^2 + 3(-1/y) + 5 = 0, which is 1/y^2 - 3/y + 5 = 0. Multiplying by y^2 results in 5y^2 - 3y + 1 = 0.

AI explanation

For the original equation x squared plus 3x plus 5 equals 0, Vieta's formulas state the sum of the roots alpha plus beta is negative 3 and the product alpha beta is 5. The new equation must have roots of negative 1 over alpha and negative 1 over beta, so the new sum is negative (alpha plus beta) divided by alpha beta, which is negative negative 3 over 5, yielding 3 over 5. The new product is 1 over alpha beta, which equals 1 over 5, so substituting these into x squared minus (sum)x plus (product) equals 0 and clearing the denominator gives 5x squared minus 3x plus 1 equals 0.

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