Question

If $\alpha$ and $\beta$ are the roots of the equation $x^2 + 5x - 7$, and $(\alpha + 1), (\beta + 1)$ are the roots of the equation $2x^2 + ax + b = 0$, what is the value of $a + b$?

• A. 0
• B. -2
• C. -16
• D. 6

Answer 1

Answer: (C)

-16

Answer 2

Let the roots of the equation $x^2 + 5x - 7 = 0$ be $\alpha$ and $\beta$. Using Vieta's formulas, we have:

• Sum of roots: $\alpha + \beta = -5$
• Product of roots: $\alpha \beta = -7$

The roots of the equation $2x^2 + ax + b = 0$ are $(\alpha + 1)$ and $(\beta + 1)$. Using Vieta's formulas for the second equation:

• Sum of roots: $(\alpha + 1) + (\beta + 1) = \alpha + \beta + 2 = -5 + 2 = -3 = -a/2$. Thus, $a = 6$.
• Product of roots: $(\alpha + 1)(\beta + 1) = \alpha \beta + \alpha + \beta + 1 = -7 - 5 + 1 = -11 = b/2$. Thus, $b = -22$.

Therefore, $a + b = 6 + (-22) = -16$.

Alternatively, substitute $x = y - 1$ into $x^2 + 5x - 7 = 0$:

$(y - 1)^2 + 5(y - 1) - 7 = 0$ $y^2 - 2y + 1 + 5y - 5 - 7 = 0$ $y^2 + 3y - 11 = 0$

The second equation is $2x^2 + ax + b = 0$, which can be written as $x^2 + (a/2)x + (b/2) = 0$. Comparing this to $y^2 + 3y - 11 = 0$, we have $a/2 = 3$ and $b/2 = -11$, so $a = 6$ and $b = -22$. Therefore, $a + b = 6 - 22 = -16$.