Multiple choice

If $\alpha$ and $\beta$ are the roots of the equation $x^2-3x+2=0$, form a equation whose roots are of $-\alpha $ and $-\beta$.

  1. $x^2+3x+2=0$
  2. $x^2-3x+2=0$
  3. $x^2+3x-2=0$

  4. None of these
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A Correct answer
Explanation

If roots are alpha and beta, the equation is x^2 - (alpha+beta)x + alpha*beta = 0. Here alpha+beta = 3 and alpha*beta = 2. New roots are -alpha and -beta. New sum = -(alpha+beta) = -3. New product = (-alpha)*(-beta) = alpha*beta = 2. New equation is x^2 - (-3)x + 2 = 0, which is x^2 + 3x + 2 = 0.

AI explanation

For the original equation x squared minus 3x plus 2 equals 0, Vieta's formulas show the sum of the roots alpha plus beta is 3 and the product alpha beta is 2. The new equation requires roots of negative alpha and negative beta, meaning its sum of roots is negative alpha minus beta, which equals negative 3, and its product is (negative alpha) times (negative beta), which equals positive 2. Applying the standard quadratic form x squared minus (sum)x plus (product) equals 0, the resulting equation is x squared plus 3x plus 2 equals 0.

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