Question

adam is using the equation \\((x)(x + 2) = 255\\) to find two consecutive odd integers with a product of 255. when adam solves the problem using the zero product property, what do those solutions represent?

two possible \first\ integers
two possible \second\ integers
the two consecutive odd integers with a product of 255
a possible \first\ integer and a possible \second\ integer

Explanation:

Define the variable and equation setup

Let \(x\) represent the first odd integer. The next consecutive odd integer is \(x + 2\). The equation modeling their product is:

$$x(x + 2) = 255$$

Analyze the solutions of the quadratic equation

Expanding and rearranging into standard form:

$$x^2 + 2x - 255 = 0$$

Factoring the quadratic equation:

$$(x + 17)(x - 15) = 0$$

Applying the zero product property yields two solutions for \(x\):

$$x = -17 \quad \text{or} \quad x = 15$$

Interpret the meaning of the solutions

Since \(x\) is defined as the first integer of the consecutive pair, the two solutions \(x = -17\) and \(x = 15\) represent the two possible values for this "first" integer.

• If the first integer is \(15\), the second is \(17\).
• If the first integer is \(-17\), the second is \(-15\).

Thus, the solutions represent two possible "first" integers.

Answer:

• two possible "first" integers (Correct answer)
• two possible "second" integers
• the two consecutive odd integers with a product of 255
• a possible "first" integer and a possible "second" integer