Question

algebra: concepts and connections ic sem b mcclendon
solving quadratic equations: factoring
finding consecutive integers
the product of two consecutive, negative integers is 182.
what equation represents the situation?
x(x + 1) = 182
what are the two numbers?
-14 and -13
-1 and 0
13 and 14

Explanation:

Step1: Set variable for negative integer

Let $x$ = first negative integer, so $x+1$ = next consecutive negative integer.

Step2: Use given product to set equation

We know $x(x+1)=182$. Rearrange to standard quadratic form:
$x^2 + x - 182 = 0$

Step3: Factor the quadratic equation

Find two numbers that multiply to $-182$ and add to $1$: $14$ and $-13$.
$(x + 14)(x - 13) = 0$

Step4: Solve for x

Set each factor equal to 0:
$x + 14 = 0 \implies x = -14$
$x - 13 = 0 \implies x = 13$
We discard $x=13$ since we need negative integers.

Step5: Find the second integer

If $x=-14$, then $x+1 = -14 + 1 = -13$

Step6: Verify the product

$(-14)(-13) = 182$, which matches the given condition.

Answer:

-14 and -13