From the January 2016 Common Core Algebra I Regents:
p is the number of people — which must be positive leaving answers (1) and (2).
What is an integer? Integer is related to the word integrity which means whole.
Real numbers means all numbers including non-integers like 1.5 — you cannot have 1.5 people
From the January 2016 Common Core Algebra I Regents
http://www.regentsprep.org/regents/math/algtrig/ate12/completesqlesson.htm
We can use the method of completing the square — see the link above
So far we have x^2 – 8x so we take 1/2 of -8 and square that to complete
(x – 4) (x – 4 ) giving x^2 – 8x + 16
trying answer (1) gives x^2 – 8x + 16 = 9 if we subtract 9 from each side we get:
x^2 – 8x + 7 = 0 close but no cigar
trying answer (2) gives x^2 – 8x + 16 = 23 if we subtract 23 from each side we get:
x^2 – 8x – 7 = 0 matches exactly!!!
OR
we can expand each answer and set it equal to zero to see which one matches the original whether on paper or on the TI-84
On the TI-84, set Y1 = x^2 – 8x – 7 and then set each answer choice to 0 and put into Y2 one by one
Try answer (1) by setting Y2 = (x – 4)^2 – 9, does it match Y1?
Try answer (2) by setting Y2 = (x – 4)^2 – 23, does it match Y1? :)
From the January 2016 Common Core Algebra I Regents:
This is really a rate of change or slope question: (change in y) / (change in x)
For answer choice (1), it took 73 years to go up 5 cents not a lot of increase per year
For answer choice (2), 14-6 = 8 cents over 14 years (1985-1971)
For answer choice (3), 24 – 14 = 10 cents over 21 years (2006 – 1985)
For answer choice (4), 35 – 24 = 11 cents over only 6 years (2012 – 2006)
(We can see that the price change has been increasing by more cents as time goes on)
So which time interval had the most change per year? Please see below
From the January 2016 Common Core Algebra I Regents
We can ‘do the FOIL’ or double distributive and/or we can use the TI-84 to find the equivalence to the given. The choices look appealing as both 3 x 10 and 2 x 15 are factor pairs of 30.
To do FOIL, we do the parentheses first in each case and then multiply by x and see if it matches the given x^3 – 13x ^2 -30x
(1) FOILing (x + 3) (x – 10) gives x^2 – 10x + 3x – 30 = x^2 -7x -30 and when multiplied by x does not look like the original
(2) FOILing (x – 3) (x – 10) gives x^2 – 10x – 3x + 30 = x^2 -13x + 30 and when multiplied by x does not look like the original
(3) FOILing (x + 2) (x – 15) gives x^2 – 15x + 2x + 30 = x^2 -13x – 30 and when multiplied by x looks familiar!
(4) FOILing (x – 2) (x + 15) gives x^2 + 15x – 2x – 30 = x^2 +13x – 30 and when multiplied by x does not look like the original
Here is the TI-84 way — looking for equivalence :)
As you can see below answer choice (1) not a match
As you can see below answer choice (2) is also not equivalent:
Onto answer choice (3), see below: it’s a keeper –= this algebra is equivalent!!!!
from the January 2016 Common Core Algebra I Regents
No solving necessary!! Just looking for the equation!!
Students need to remember how to find the area of a rectangle: length by width
What was the original area of the patio? The original area was 120 sq ft (10 x 12)
What is 50% more than 120? Add 1/2 of 120 which is 60 making the new area 120 + 60 = 180 leaving answers (2) and (3).
Since we are adding x amount to each dimension 10 becomes 10 + x and 12 goes to 12 + x.
Those two multiplied together = 180 and voila there it is!
Math Confidence by Robin the Math Lady 