The partial factorization of x2 – x – 12 is modeled with algebra tiles. An algebra tile configuration. 4 tiles are in the Factor 1 spot: 1 is labeled + x, 3 are labeled +. 1 tile is in the Factor 2 spot and is labeled + x. 20 tiles are in the Product spot: 1 is labeled + x squared, 3 are labeled + x, the 4 tiles below + x squared are labeled negative x, and the 12 tiles below the + x tiles are labeled negative.
Which unit tiles are needed to complete the factorization?

x² – x – 12 = (x – 4)(x + 3)

Step-by-step explanation:

Identify two numbers that add to -1 and multiply to -12, let's call them p and q.

So ax² + bx + c = (x + p)(x + q)

pq = c

p + q = b.

It is easier to find these numbers by finding factors of -12.

This can be done by splitting the number up until all the numbers are prime.

-12 → 6 × -2 or -6 × 2 → -(3 × 2 × 2)

There can only be two numbers so the only options we have are 6 and -2, -6 and 2, 3, and -4, or -3 and 4.

We can eliminate them by adding them up.

6 + -2 = 4 ≠ -1 so that can't be it.

-6 + 2 = -4 ≠ -1 so that can't be it either.

-3 + 4 = 1 ≠ -1

therefore p and q are 3 and -4 because 3 + -4 = -1.

so x² – x – 12 = (x – 4)(x + 3)

p = -4, and q = 3.

(x – 4)(x + 3) = x(x + 3) – 4(x + 3) = x² + 3x – 4x + 12 = x² – x – 12

how many players are there?

step-by-step explanation:

- "runner b is moving at a slower rate."

- "the graphs shown are graphs of distance(km) against time(min).

the slopes of this graphs are are given by:

slope = (change in y)/(change in x)

= (change in distance)/(change in time)

this is the formula for finding the speed. so, the slope of these graphs represent the speed of the runner.

taking each division in the graph to represent 1 on both axis:

runner a ⇒ speed = (2-0)/(1-0)

= 2/1

= 2 km/min

runner b ⇒ speed = (3-0)/(3-0)

= 3/3

= 1 km/min

this shows that   runner b is moving at a slower rate."