Question

1. Multiply and simplify if possible.
(2x+4)(x-2)

2. The sum of the reciprocals of two consecutive even integers is 9/40. This can be represented by the equation shown. 1/x + 1/(x+2) = 9/40. Use the rational equation to determine the integers.

Answer 1

1. To multiply (2x+4)(x-2), we first distribute the (2x) and (4) to each term inside the parentheses:
2x * x = 2x^2
2x * (-2) = -4x
4 * x = 4
4 * (-2) = -8

Now we combine the terms:
2x^2 - 4x + 4 - 8

Simplify by combining like terms:
2x^2 - 4x - 4

Therefore, (2x+4)(x-2) simplifies to 2x^2 - 4x - 4.

2. The equation 1/x + 1/(x+2) = 9/40 can be simplified to (x+2 + x) / (x * (x+2)) = 9/40, which further simplifies to (2x + 2) / (x^2 + 2x) = 9/40.

Now, cross multiply to get:
40(2x + 2) = 9(x^2 + 2x)
80x + 80 = 9x^2 + 18x
0 = 9x^2 - 62x - 80

Now, solve for x by factoring or using the quadratic formula. The solutions will give you the two consecutive even integers.