Multiple choice

If a and b are the roots of the quadratic equation $\displaystyle { 6x }^{ 2 }-x-2=0$ from an equation whose roots are $\displaystyle { a }^{ 2 }$ and $\displaystyle { b }^{ 2 }$ ?

  1. $\displaystyle { 36x }^{ 2 }-25x+4=0$
  2. $\displaystyle { 36x }^{ 2 }+25x+4=0$
  3. $\displaystyle { 6x }^{ 2 }-25x+4=0$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For 6x^2 - x - 2 = 0, a+b = 1/6 and ab = -1/3. We want an equation with roots a^2 and b^2. Sum = a^2 + b^2 = (a+b)^2 - 2ab = (1/6)^2 - 2(-1/3) = 1/36 + 2/3 = 1/36 + 24/36 = 25/36. Product = (ab)^2 = (-1/3)^2 = 1/9 = 4/36. The equation is x^2 - (25/36)x + 4/36 = 0, which is 36x^2 - 25x + 4 = 0.

AI explanation

By Vieta's formulas for the equation 6x^2 - x - 2 = 0, the sum of the roots a + b equals 1/6 and the product ab equals -2/6 = -1/3. For the new equation with roots a^2 and b^2, the required sum is a^2 + b^2 = (a + b)^2 - 2ab = (1/6)^2 - 2(-1/3) = 1/36 + 2/3 = 25/36, and the required product is a^2 * b^2 = (ab)^2 = (-1/3)^2 = 1/9. The new quadratic equation is therefore x^2 - (sum)x + (product) = 0, which becomes x^2 - (25/36)x + 1/9 = 0. Multiplying the entire equation by 36 gives the final result: 36x^2 - 25x + 4 = 0.

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