Quadratic Equations Quiz 1
Key
Name _________________________________________
Period _____ Date ________
Directions:
Read each question fully and carefully. Answer each question completely.
You must show your work. Each question is worth five (5) points.
1. Solve by completing the square:
x 2  6x  2  0
𝑥 2 − 6𝑥 = −2
𝑥 2 − 6𝑥 + 9 = 7
𝑥−3 2 =7
𝑥−3=± 7
𝑥 = 3± 7
Subtract 2
Add −6/2 2 = 9
Write as perfect square
Square root
Solution
2. Solve using the quadratic formula:
4 x2  2 x  1  0
𝑎=4
𝑏=2
𝑥=
𝑥=
;𝑏± 𝑏 2 ;4𝑎𝑐
2𝑎
;2± 4;16
8
1
𝑥 = −4 ±
3
4
𝑐=1
=
=
;2± 22 ;4∙4∙1
Quadratic Formula
2∙4
;2± ;12
8
=
;2±𝑖 4 3
8
𝑖
Solution
3. Solve the following problem. Approximate answers involving radicals to the
nearest hundredth.
Find the dimensions of a rectangle whose perimeter is 10 cm and whose area
is 3 cm2.
𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 2𝑤 + 2ℎ = 10 𝑐𝑚
𝑎𝑟𝑒𝑎 = 𝑤 ∙ ℎ = 3 𝑐𝑚2
ℎ =5−𝑤
solve perimeter equation for h
h
𝑤 5−𝑤 =3
substitute into area equation
2
5𝑤 − 𝑤 = 3
simplify
w
𝑤 2 − 5𝑤 + 3 = 0
standard form
𝑎=1
𝑏 = −5
𝑐=3
𝑤=
𝑤=
; ;5 ±
;5 2 ;4∙1∙3
Quadratic Formula
2∙1
5 ± 25 − 12 5 ± 13
=
2
2
Dimensions:
0.70 cm  4.30 cm
-1-
Quadratic Equations Quiz 1
Key
Name _________________________________________
Period _____ Date ________
Directions:
Read each question fully and carefully. Answer each question completely.
You must show your work. Each question is worth five (5) points.
1. Solve by completing the square:
x 2  5x  3
𝑥 2 − 5𝑥 +
5 2
𝑥−2
25
=
5
37
Add −5/2
4
2
=
25
4
Write as perfect square
4
37
𝑥−2=±
𝑥=
=
4
37
Square root
4
5± 37
Solution
2
2. Solve using the quadratic formula:
3x 2  9 x  2
3𝑥 2 + 9𝑥 + 2 = 0
𝑎=3
𝑥=
𝑥=
Add 2
𝑏=9
;𝑏± 𝑏 2 ;4𝑎𝑐
2𝑎
;9± 81;24
6
𝑐=2
=
=
;9± 92 ;4∙3∙2
Quadratic Formula
2∙3
;9± 57
Solution
6
3. Solve the following problem. Approximate answers involving radicals to the
nearest hundredth.
A sidewalk of uniform width has area 180 ft2 and surrounds a flower bed that
is 11 ft wide and 13 ft long. Find the width of the sidewalk.
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑆𝑖𝑑𝑒𝑤𝑎𝑙𝑘 = 2 𝐴𝑟𝑒𝑎 𝑜𝑓 𝐵𝑜𝑡𝑡𝑜𝑚 + 2 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑆𝑖𝑑𝑒
w
13
11
Bottom
Side
𝐴 = 2 ∙ 11𝑤 + 2 ∙ 13 + 2𝑤 𝑤 = 180
Area of Sidewalk
22𝑤 + 26𝑤 + 4𝑤 2 = 180
4𝑤 2 + 48𝑤 − 180 = 0
Standard Form
𝑤 2 + 12𝑤 − 45 = 0
Simplify
𝑤=
;12± 122 ;4∙1∙ ;45
Quadratic Formula
2∙1
𝑤=
;12± 144:180
𝑤=
;12±18
2
2
=
;12± 324
= −15, 3
𝑤 = 3 𝑓𝑡
2
Solution
Width cannot be negative
-2-
Quadratic Equations Quiz 1
Key
Name _________________________________________
Period _____ Date ________
Directions:
Read each question fully and carefully. Answer each question completely.
You must show your work. Each question is worth five (5) points.
1. Solve by completing the square:
x2  8x  3  0
𝑥 2 − 8𝑥 = −3
𝑥 2 − 8𝑥 + 16 = 13
𝑥 − 4 2 = 13
𝑥 − 4 = ± 13
𝑥 = 4 ± 13
Subtract 3
Add −8/2 2 = 16
Write as perfect square
Square root
Solution
2. Solve using the quadratic formula:
3x 2  3x  1  0
𝑎=3
𝑏=3
𝑥=
𝑥=
;𝑏± 𝑏 2 ;4𝑎𝑐
2𝑎
;3± 9;12
6
1
𝑥 = −2 ±
3
6
=
=
𝑐=1
;3± 32 ;4∙3∙1
Quadratic Formula
2∙3
;3± ;3
6
=
;3±𝑖 3
6
𝑖
Solution
3. Solve the following problem. Approximate answers involving radicals to the
nearest hundredth.
Find the dimensions of a rectangle whose perimeter is 42 cm and whose area
is 20 cm2.
h
w
𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 2𝑤 + 2ℎ = 42 𝑐𝑚
𝑎𝑟𝑒𝑎 = 𝑤 ∙ ℎ = 20 𝑐𝑚2
ℎ = 21 − 𝑤
𝑤 21 − 𝑤 = 20
21𝑤 − 𝑤 2 = 20
𝑤 2 − 21𝑤 + 20 = 0
𝑎=1
𝑏 = −21
𝑐 = 20
𝑤=
𝑤=
; ;21 ±
;21 2 ;4∙1∙20
2∙1
solve perimeter equation for h
substitute into area equation
simplify
standard form
Quadratic Formula
21 ± 441 − 80 21 ± 361 21 ± 19
=
=
2
2
2
Dimensions:
1.00 cm  20.00 cm
-3-
Quadratic Equations Quiz 1
Key
Name _________________________________________
Period _____ Date ________
Directions:
Read each question fully and carefully. Answer each question completely.
You must show your work. Each question is worth five (5) points.
1. Solve by completing the square:
x2  5x  3
25
𝑥 2 + 5𝑥 +
5 2
𝑥+2
=
5
37
Add 5/2
4
2
=
25
4
Write as perfect square
4
37
𝑥+2=±
𝑥=
=
4
37
Square root
4
;5± 37
Solution
2
2. Solve using the quadratic formula:
4 x 2  7 x  2
4𝑥 2 − 7𝑥 + 2 = 0
𝑎=4
𝑥=
𝑥=
Add 2
𝑏 = −7
;𝑏± 𝑏 2 ;4𝑎𝑐
2𝑎
7± 49;32
8
=
=
𝑐=2
;7 2 ;4∙4∙2
7±
Quadratic Formula
2∙4
7± 17
Solution
8
3. Solve the following problem. Approximate answers involving radicals to the
nearest hundredth.
A sidewalk of uniform width has area 352 ft2 and surrounds a flower bed that
is 17 ft wide and 19 ft long. Find the width of the sidewalk.
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑆𝑖𝑑𝑒𝑤𝑎𝑙𝑘 = 2 𝐴𝑟𝑒𝑎 𝑜𝑓 𝐵𝑜𝑡𝑡𝑜𝑚 + 2 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑆𝑖𝑑𝑒
w
19
17
Bottom
Side
𝐴 = 2 ∙ 17𝑤 + 2 ∙ 19 + 2𝑤 𝑤 = 352
Area of Sidewalk
34𝑤 + 38𝑤 + 4𝑤 2 = 352
4𝑤 2 + 72𝑤 − 352 = 0
Standard Form
𝑤 2 + 18𝑤 − 88 = 0
Simplify
𝑤=
;18± 182 ;4∙1∙ ;88
Quadratic Formula
2∙1
𝑤=
;18± 324:352
𝑤=
;18±26
2
2
=
;18± 676
= −22, 4
𝑤 = 4 𝑓𝑡
2
Solution
Width cannot be negative
-4-