Multiple choice

The equation formed by the increasing each root of $ax^2+bx+c=0$ by $1$, is $2x^2+8x+2=0$, then

  1. $a+b=0$
  2. $b+c=0$
  3. $b=c$
  4. $a=b$
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C Correct answer
AI explanation

If the roots of ax^2 + bx + c = 0 are increased by 1, the new roots are alpha + 1 and beta + 1, which satisfy the equation 2x^2 + 8x + 2 = 0. Dividing by 2 gives x^2 + 4x + 1 = 0, so the sum of the new roots is -4 and their product is 1. We know (alpha + 1) + (beta + 1) = alpha + beta + 2 = -b/a + 2, so -b/a + 2 = -4, meaning -b/a = -6 and b = 6a. For the product, (alpha + 1)(beta + 1) = alpha*beta + (alpha + beta) + 1 = c/a - b/a + 1 = 1, which simplifies to c/a - 6 = 0, so c = 6a. Since both b and c equal 6a, b equals c. The result is b = c.

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