Final answer:
To find the consecutive even integers whose product is 224, set up the equation x(x+2) = 224, solve the quadratic equation to get x=14 or x=-16, discard the negative value, and add 2 to 14 to find the next integer, leading to the integers 14 and 16.
Step-by-step explanation:
The question involves finding two consecutive even integers whose product is 224. To solve this, we can express the consecutive even integers as x (the smaller even integer) and x+2 (the next even integer). The equation formed is x(x+2) = 224. Solving it:
- Step 1: Expand the equation to get x^2 + 2x = 224.
- Step 2: Rearrange the equation to a standard quadratic form: x^2 + 2x - 224 = 0.
- Step 3: Factor the quadratic equation to find the integer solutions. The factors of 224 that differ by 2 are 14 and 16.
- Step 4: So the equation factors as (x - 14)(x + 16) = 0. This gives us two possible values for x: 14 or -16.
- Step 5: Neglect the negative value as we are looking for positive integers, so x = 14 is the smaller integer.
- Step 6: The next consecutive even integer will be 14+2 = 16.
Therefore, the two consecutive even integers whose product is 224 are 14 and 16, which corresponds to option A.