Multiple choice

If $\alpha$ and $\beta$ are the roots of the equation $2x^2-3x-5=0$, then form a quadratic equation whose roots are $\alpha^2 $ and $\beta^2$.

  1. $4x^2-29x+25=0$
  2. $4x^2+29+25=0$
  3. $4x^2-29-25=0$

  4. None of these
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A Correct answer
Explanation

For 2x^2 - 3x - 5 = 0, alpha + beta = 3/2 and alpha * beta = -5/2. The new roots are alpha^2 and beta^2. Their sum is (alpha + beta)^2 - 2(alpha * beta) = (3/2)^2 - 2(-5/2) = 9/4 + 5 = 29/4. Their product is (alpha * beta)^2 = 25/4. The equation is x^2 - (29/4)x + 25/4 = 0, or 4x^2 - 29x + 25 = 0.

AI explanation

For the equation 2x^2-3x-5=0, the sum of roots a+b = 3/2 and the product ab = -5/2. The new roots are a^2 and b^2, so the new sum is a^2+b^2 = (a+b)^2-2ab = (3/2)^2 - 2*(-5/2) = 9/4 + 5 = 29/4. The new product is (ab)^2 = (-5/2)^2 = 25/4. The required equation is x^2 - (29/4)x + 25/4 = 0, which simplifies to 4x^2-29x+25=0.

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