Multiple choice

Equation $x^2+5x-7 = 0$ has roots a and b. Equation $2x^2+px+q = 0 $ has roots $a + 1$ and $b + 1$. Find $p + q$.

  1. 6

  2. 0

  3. -16

  4. 2
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For x^2 + 5x - 7 = 0, a+b = -5 and ab = -7. The new roots are a+1 and b+1. Sum of new roots = a+b+2 = -5+2 = -3. Product of new roots = (a+1)(b+1) = ab + a+b + 1 = -7 - 5 + 1 = -11. The new equation is x^2 - (sum)x + (product) = x^2 + 3x - 11 = 0. Multiplying by 2 to match 2x^2 + px + q = 0 gives 2x^2 + 6x - 22 = 0. Thus p=6, q=-22. p+q = 6 - 22 = -16.

AI explanation

Using Vieta's formulas for x^2 + 5x - 7 = 0, the sum of the roots is a + b = -5 and the product is ab = -7. For the new equation with roots a + 1 and b + 1, the sum of the roots is (a + 1) + (b + 1) = (a + b) + 2, which equals -5 + 2 = -3. Since the new equation is 2x^2 + px + q = 0, the sum of the roots equals -p/2, so -p/2 = -3, giving p = 6. The product of the new roots is (a + 1)(b + 1) = ab + a + b + 1, which equals -7 + (-5) + 1 = -11. The product of the roots also equals q/2, so q/2 = -11, giving q = -22. Therefore, p + q equals 6 + (-22), which is -16.

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