Completing The Square
(CTS)
Aricka Smith
Secondary Math Education
I can do quadratics
in my head!!
" BM 13901 #7 reads
(Aaboe's
translation of
Neugebauer's
German
Translation)
Construction of CTS
" I have added seven times the side of my square to
eleven times its area and it is 6;15.
!
11X
X
7
= 6
" You take 7 and 11. You multiply 11 by 6;15 and it is
1,8;45. You halve 7 [and obtain 3;30]. You multiply 3;30
and 3;30. You add [the result] 12;15 to 1,8;45 and [the
result]
11X
11X
7
6
(11)
" You take 7 and 11. You multiply 11 by 6;15 and it is
1,8;45. You halve 7 [and obtain 3;30]. You multiply 3;30
and 3;30. You add [the result] 12;15 to 1,8;45 and [the
result]
11X
11X
3
3
68
" You take 7 and 11. You multiply 11 by 6;15 and it is
1,8;45. You halve 7 [and obtain 3;30]. You multiply 3;30
and 3;30. You add [the result] 12;15 to 1,8;45 and [the
result]
11X
11X
3
3
68
" You take 7 and 11. You multiply 11 by 6;15 and it is
1,8;45. You halve 7 [and obtain 3;30]. You multiply 3;30
and 3;30. You add [the result] 12;15 to 1,8;45
11X
3
11X
68
+12
3
12
" You take 7 and 11. You multiply 11 by 6;15 and it is
1,8;45. You halve 7 [and obtain 3;30]. You multiply 3;30
and 3;30. You add [the result] 12;15 to 1,8;45
11X
3
11X
3
81
12
" and [the result] 1,21 has 9 as it's square root. You
subtract 3;30, which you multiplied by itself, from 9 and
you have 5;30.
" 11x +3
=9
" 11x
=5
" The reciprocal of 11 does not divide. What shall I
multiply by 11 so that 5;30 results?
" 0;30 is the factor. 0;30 is the side of the square
" X=
Connection to Quadratics in my
Head:
" Divide b by 2
" Square that
X
number
" Subtract c from
that number
" Then roots are
X
-C
How CTS connects to
Vertex Form
" Standard form: y= 2x2 + 8x – 7
x
x
4
How CTS connects to
Vertex Form
" Standard form: y= 2x2 + 8x – 7
x
x
2
2
How CTS connects to
Vertex Form
" Standard form: y= 2x2 + 8x – 7
x
x
2
2
How CTS connects to
Vertex Form
" Standard form: y= 2x2 + 8x – 7
x
2
x
+4
2
4
Vertex Form
"(x+2)2 =
+4
"(x+2)2-4=
"2(x+2)2-4=y+7
"2(x+2)2-15=y
"Vertex: (-2, -15)
How CTS connects elsewhere
Connection to Conics
Stewart, Redlin, Watson Precalculus
Why this interested me
" Old method; Connections
" Guidance; Visual