Multiple choice

lf each root of the equation $3\mathrm{x}^{2} -7x +4=0$ is increased by $2$, then the resulting equation is

  1. $3\mathrm{x}^{2}-19\mathrm{x}+30=0$
  2. $3\mathrm{x}^{2}+5\mathrm{x}+2=0$
  3. $3\mathrm{x}^{2}-19\mathrm{x}+2=0$
  4. $3\mathrm{x}^{2}-19\mathrm{x}+20=0$
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A Correct answer
Explanation

If roots are increased by 2, replace x with (x-2). 3(x-2)^2 - 7(x-2) + 4 = 0. 3(x^2 - 4x + 4) - 7x + 14 + 4 = 0. 3x^2 - 12x + 12 - 7x + 18 = 0. 3x^2 - 19x + 30 = 0.

AI explanation

First, find the sum and product of the roots of 3x^2 - 7x + 4 = 0, which are 7/3 and 4/3. When each root is increased by 2, the new sum of the roots becomes (7/3) + 2(2) = 19/3, and the new product becomes (4/3) + 2(7/3) + 2(2) = 10. The new quadratic equation with these roots is x^2 - (Sum)x + Product = 0, so we have x^2 - (19/3)x + 10 = 0. Multiplying the entire equation by 3 yields the resulting equation 3x^2 - 19x + 30 = 0.

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