Solutions
Algebra: 8.1.2 Factoring w/Generic Rectangles Name ______________________
Block _____ Date __________
Bell Work 1/7
a.
Factor each generic rectangle. Write the area of each as a product = simplified sum.
b.
+5 5x 25
c.
-2 -2x -4
x x2 5x
x +5
x x2 2x
x +2
(𝒙 + 𝟓)𝟐 = 𝒙𝟐 + 𝟏𝟏𝟏 + 𝟐𝟐
(𝒙 + 𝟐)(𝒙 − 𝟐) = 𝒙𝟐 − 𝟒
-5 -5x -35
2x 2x2 14x
x +7
(𝒙 + 𝟕)(𝟐𝟐 − 𝟓) = 𝟐𝒙𝟐 + 𝟗𝟗 − 𝟑𝟑
𝟏
A. Graph the quadratic parent function 𝒚 = 𝒙𝟐 and 𝒚 = 𝟐 𝒙𝟐 − 𝟐 on the same grid.
1
a. What are the root(s), vertex and y-intercept(s) for 𝑦 = 2 𝑥 2 − 2?
Roots = -2 and 2
Vertex = (0, -2)
y-intercept = -2
1
b. How does the graph of 𝑦 = 2 𝑥 2 − 2 compare to the parent function?
The graph is 2 units lower and
half as steep
B. Factor and graph each quadratic equation. What are the root(s), vertex and y-intercept(s) for each?
a.
y = x2 + 4x +3 = (𝑥 + 3)(𝑥 + 1)
+3 +3𝑥
x 𝑥
2
x
3
y = x2 − 6x + 9 = (𝑥 − 3)2
b.
−3 −3𝑥
+𝑥
x 𝑥2
+1
x
Roots = -3 and -1
Vertex = (-2, -1)
y-intercept = 3
9
−3𝑥
−3
Root = 3
Vertex = (3, 0)
y-intercept = 9
8-13. Factor the generic rectangle. Write the area as a product = simplified sum.
−7 −35x 14
2x 10x2 −4x
5x −2
(𝟓𝟓 − 𝟐)(𝟐𝟐 − 𝟕) = 𝟏𝟏𝒙𝟐 − 𝟑𝟑𝟑 + 𝟏𝟒
8-14. FACTORING QUADRATIC EXPRESSIONS
Factor each trinomial. Model each with algebra tiles and a generic rectangle.
a.
b.
c.
2𝑥 2 + 5𝑥 + 3 = (2𝑥 + 1)(𝑥 + 1)
+1 2𝑥
+3
2x
+1
x 2𝑥 2
3𝑥 2 + 10𝑥 + 8 = (3𝑥 + 4)(𝑥 + 2)
+2 6𝑥
2𝑥 2 + 7𝑥 + 6 = (2𝑥 + 3)(𝑥 + 2)
+3 +4𝑥 +6
x 3𝑥 2
3x
𝑥 2 − 𝑥 − 2 = (𝑥 + 1)(𝑥 − 2)
+1 +𝑥
x 𝑥2
x
e.
3x2 + 10x – 8 = (𝑥 + 4)(3𝑥 − 2)
+8
4𝑥
+4
2x 2𝑥 2 +3𝑥
x
d.
3𝑥
+2
−2
−2𝑥
−2
+4 12𝑥 −8
x 3𝑥 2 −2𝑥
3x
−2
8-15 Factor the quadratic equation using a generic rectangle only:
y = 6x2 + 17x + 12
𝒚 = (𝟑𝒙 + 𝟒)(𝟐𝟐 + 𝟑)
+3 9𝑥
+12
3x
+4
2x 6𝑥 2
8𝑥
8-16. Factor the following quadratic expressions, if possible. If an expression cannot be factored,
justify your conclusion. Be sure to look for a common factor first!
+6 6𝑥
a. x + 9x + 18 = (𝒙 + 𝟔)(𝒙 + 𝟑)
2
x 𝑥2
x
b. 4x + 17x − 15 = (𝟒𝟒 − 𝟑)(𝒙 + 𝟓)
2
+18
3𝑥
+3
+5 20𝑥 −15
x 4𝑥 2 −3𝑥
4x
c. 8x − 16x + 6 = 2(4𝑥2 − 8𝑥 + 3) = 𝟐(𝟐𝟐 − 𝟏)(𝟐𝟐 − 𝟑)
2
2
d. 3x + 5x – 3
3𝑥 2
−3
−3 −6𝑥 +3
−3
2x
2x 4𝑥 2 −2𝑥
−1
3x2 + 5x – 3 is prime (cannot be factored). No terms exist that have a product of -9x2 and a sum of 5x.
8-17. Factor the following quadratics, if possible. (Look for a common factor first!)
a. x2 − 4x – 12 = (𝒙 − 𝟔)(𝒙 + 𝟐)
b. 4x + 4x + 1 = (𝟐𝒙 + 𝟏
2
−6 −6𝑥 −12
)𝟐
x 𝑥2
x
2𝑥
+2
c. 2x2 − 9x – 5 = (𝒙 − 𝟓)(𝟐𝒙 + 𝟏)
+1 2𝑥
4𝑥 2
2𝑥
d. 2x2 + 12x + 10 = 2(𝑥 2 + 6𝑥 + 5) = 𝟐(𝒙 + 𝟓)(𝒙 + 𝟏)
2𝑥
+5
x
5𝑥
𝑥2
𝑥
1
2𝑥
−5 −10𝑥 −5
+1
x 2𝑥 2
2𝑥
5
1𝑥
+1
8-18. For each rule represented below, state the x- and y-intercepts, if possible.
x-intercepts = -3 and -1
y-intercept = 2
x-intercepts = -1 and 3
y-intercept = -3
5𝑥 − 2(0) = 40
5(0) − 2𝑦 = 40
𝑥=8
𝑦 = −20
5𝑥 = 40
x-intercept = 2, no y-intercepts
−2𝑦 = 40
x-intercept = 8 and y-intercept = -20
1𝑥
+1
1 𝑥
𝑦=� �
2
𝑦 = −2𝑥 − 5.5
50(0.92)5 ≈ $32.95
𝑥 + 2𝑥 − 3 = 15
3𝑥 = 18
𝑥=6
3𝑥 + 2 = 𝑦
𝑦 = 2(6) − 3
𝑦=9
6𝑥 = 4 − 2(3𝑥 + 2)
6𝑥 = 4 − 6𝑥 − 4
6𝑥 = −6𝑥
12𝑥 = 0
𝑥=0
28𝑥 = 5𝑥 − 10
23𝑥 = −10
−𝟏𝟏
𝒙=
𝟐𝟐
Δ𝑦 100 1
=
=
Δ𝑥 400 4
3(0) + 2 = 𝑦
2=𝑦
−6𝑏 + 21 = −6𝑏 + 21
True
𝒃 is any real number
𝑦 = 𝑚𝑚 + 𝑏
1
200 = (−800) + 𝑏
4
200 = −200 + 𝑏
400 = 𝑏
(𝟔, 𝟗)
𝒚=
𝟏
𝒙 + 𝟒𝟒𝟒
𝟒
(𝟎, 𝟐)
6 − 2𝑐 + 6 = 12
−2𝑐 + 12 = 12
−2𝑐 = 0
𝒄=𝟎
EOC Practice:
5 hours is 300 minutes
(300, 16000)
13000
600
300 min ∙
10 𝑔𝑔𝑔𝑔𝑔𝑔𝑔
1 𝑚𝑚𝑚
3000 gallons
=
The number of gallons
started 3000 less than
16000 or 13000 gallons
10 𝑔𝑔𝑔𝑔𝑔𝑔𝑔 60 𝑚𝑚𝑚
∙
1 𝑚𝑚𝑚
1 ℎ𝑟
=
600 𝑔𝑔𝑔𝑔𝑔𝑔𝑔
1 ℎ𝑜𝑜𝑜