Question

The roots of $y^2 + 5y - 11 = 0$ are $a$ and $b$. Find $(a + 3)(b + 3).$

Answer 1

Since a and b are roots of y² + 5y - 11, we have

y² + 5y - 11 = (y - a) (y - b)

and expanding the right side gives

y² + 5y - 11 = y² - (a + b) y + ab

so that

a + b = -5

ab = -11

Now,

(a + 3) (b + 3) = ab + 3 (a + b) + 9

(a + 3) (b + 3) = -11 + 3 (-5) + 9

(a + 3) (b + 3) = -17