Question

two negative integers are 8 units apart on the number line and have a product of 308. which equation could be used to determine x, the smaller negative integer? \\(\bigcirc\\) \\(x^2 + 8x - 308 = 0\\) \\(\bigcirc\\) \\(x^2 - 8x + 308 = 0\\) \\(\bigcirc\\) \\(x^2 + 8x + 308 = 0\\) \\(\bigcirc\\) \\(x^2 - 8x - 308 = 0\\)

Explanation:

Step1: Define the larger integer

Let \( x \) be the smaller negative integer. Since the two negative integers are 8 units apart, the larger negative integer is \( x + 8 \) (because on the number line, a larger negative integer is closer to zero, so if \( x \) is smaller, adding 8 gives the larger one).

Step2: Set up the product equation

The product of the two integers is 308. So, we have the equation:

$$ x(x + 8) = 308 $$

Step3: Expand and simplify the equation

Expanding the left - hand side:

$$ x^{2}+8x = 308 $$

Subtract 308 from both sides to get the quadratic equation in standard form:

$$ x^{2}+8x - 308=0 $$

Answer:

\( x^{2}+8x - 308 = 0 \) (the first option)