Question

the product of two consecutive even integers is 224. find the integers.

smaller integer =

larger integer =

Explanation:

Define variables for consecutive even integers

Using the Consecutive Integer Word Problems knowledge point
Let the smaller even integer be \(x\).
The next consecutive even integer is \(x + 2\).

Set up the quadratic equation

Using the Translating Word Problems to Quadratic Equations knowledge point

$$ x(x + 2) = 224 $$

$$ x^2 + 2x - 224 = 0 $$

Solve the quadratic equation

Using the Solving Quadratic Equations knowledge point

$$ (x + 16)(x - 14) = 0 $$

$$ x = -16 \quad \text{or} \quad x = 14 $$

Determine the integer pairs

Using the Integer Word Problems knowledge point
If \(x = 14\):
Smaller integer = \(14\), Larger integer = \(16\).

If \(x = -16\):
Smaller integer = \(-16\), Larger integer = \(-14\).

Since standard school algebra problems typically seek the positive integer solution unless specified otherwise, we provide the positive pair \(14\) and \(16\).

Answer:

The product of two consecutive even integers is 224. Find the integers.

smaller integer = <blank>14</blank>

larger integer = <blank>16</blank>