Question

one integer is 4 more than another. their product is 165. find the integers.

smaller integer =
larger integer =

Explanation:

Define variables for the integers

Using the Integer Word Problems knowledge point
Let the smaller integer be \(x\).
Since one integer is 4 more than another, the larger integer is \(x + 4\).

Set up the quadratic equation

Using the Integer Word Problems knowledge point

$$ x(x + 4) = 165 $$

$$ x^2 + 4x - 165 = 0 $$

Solve the quadratic equation

Using the Solving Quadratic Equations knowledge point

$$
LATEXBLOCK0
$$

Determine the pairs of integers

Using the Integer Word Problems knowledge point
If the smaller integer is \(x = 11\), the larger integer is:

$$ x + 4 = 11 + 4 = 15 $$

If the smaller integer is \(x = -15\), the larger integer is:

$$ x + 4 = -15 + 4 = -11 $$

Both pairs \((11, 15)\) and \((-15, -11)\) are valid integer solutions. Typically, standard algebra curricula accept either pair, but the positive pair is most commonly expected. We will provide both possibilities.

Answer:

One integer is 4 more than another. Their product is 165. Find the integers.

smaller integer = <blank>\(11\)</blank> or <blank>\(-15\)</blank>
larger integer = <blank>\(15\)</blank> or <blank>\(-11\)</blank>