math

1. Quadratic equations in
in this form:
ax 2 bx c 0, a 0
1) The Square Root Property
if u 2
d , then u
d
2) Completing the Square
a) Start with form: x 2
bx
b
b) Half b and square it:
2
c) Add to x 2
bx :
2
2
b
x 2 bx
2
d) It is now a perfect square:
x2
bx
b
2
2
x
b
2
2

2. For Quadratic equations in this
form: ax 2 bx c 0, a 0
Use this rule
If A ê B 0,
Then A 0 or B
0
Ch6 Review # 82,84,88,92
#82 x x 12
0
0
x
0 or x 12
#84 x2 5x 14
0
x 7
x 2
x 7 0 x 2
x
7 or x 2
0
0
0

3. Ch6 Review # 82,84,88,92
#88
x 3 x 2
x2
50
2x 3x 6 50
x
2
x 56
0
0
x 8 x 7
x 8 0 x 7
0
x
8
or x 7
0
Rule
If A ê B 0,
Then A 0 or B
0
Use this rule for
Quadratic equations
in in this form:
ax 2
bx
c
0, a
0

4. Rule
#92 3x2
3x 2
22x 7
22x 7
0
0
3x 1 x 7
3x 1 0 x 7
0
x
1
or x
3
7
If A ê B 0,
Then A 0 or B
0
Use this rule for
Quadratic equations
in in this form:
ax 2
bx
c
0, a
0

5. 1) The Square Root Property
if u 2
d,
then u
d and u
d
if u 2
d , then u
Quadratic equations in
in this form:
ax 2 bx c 0, a 0
d
2) Completing the Square
2
2
b
b
2
x bx
x
2
2
3) Solving by Completing the Square

6. 1) The Square Root Property
if u 2
d , then u
d
11.1 #2,6,9,15,22
# 2 5x 2
x2
x
2
2
d , then u
d
2) Completing the
Square
u2
x2
d ?
4 then u
x
x
if u 2
20
20
5
4
Quadratic equations in
in this form:
ax 2 bx c 0, a 0
1) The Square Root
Property
d
bx
b
2
2
x
3) Solving by
Completing the
Square
b
2
2

7. Quadratic equations in
in this form:
ax 2 bx c 0, a 0
1) The Square Root
Property
1) The Square Root Property
if u 2
d , then u
d
11.1 #2,6,9,15,22
# 6 4x
x
2
2
x
x
if u 2
49
49
4
49
4
7
2
u2
d
2) Completing the
Square
d ?
x2
then u
d , then u
d
bx
b
2
2
x
3) Solving by
Completing the
Square
b
2
2

8. Quadratic equations in
in this form:
ax 2 bx c 0, a 0
1) The Square Root
Property
1) The Square Root Property
if u 2 d , then u
11.1 #2,6,9,15,22
# 9 25x2 16
x
x
x
x
x
2
d
0
16
u2 d ?
25
16
then u
25
16
1
25
16
25
4
i
5
1
if u 2
d
d , then u
d
2) Completing the
Square
x2
bx
b
2
2
x
3) Solving by
Completing the
Square
b
2
2

9. Quadratic equations in
in this form:
ax 2 bx c 0, a 0
1) The Square Root
Property
1) The Square Root Property
if u 2
d , then u
d
11.1 #2,6,9,15,22
#15 2 x+2
x 2
2
2
16
u2
8
if u 2
x
x
8
d ?
2
then u
4 2
2 2 2
d
2) Completing the
Square
x2
x 2
d , then u
d
bx
b
2
2
x
3) Solving by
Completing the
Square
b
2
2

10. Quadratic equations in
in this form:
ax 2 bx c 0, a 0
1) The Square Root
Property
1) The Square Root Property
if u 2
d , then u
d
11.1 #2,6,9,15,22
# 22 x
2
6x 9
x 3
x 3
x
x
49
49
49
u
2
x2
d ?
then u
d
3 7
3 7
x
10
4,10
d , then u
d
2) Completing the
Square
49
x 3 x 3
2
if u 2
or x 3 7
4
bx
b
2
2
x
3) Solving by
Completing the
Square
b
2
2

11. Quadratic equations in
in this form:
ax 2 bx c 0, a 0
1) The Square Root
Property
2) Completing the Square
When we can’t solve by factoring, we
can complete the square and use the
square root property to solve
if u 2
How to complete the square:
x
x2
b
2
x
2
bx
b
2
x
b
2
bx
b
2
2
x
2
3) Solving by
Completing the
Square
d) It is now a perfect square:
2
d
2) Completing the
Square
2
bx
2
b
b) Half b and square it:
2
c) Add to x 2 bx : x 2 bx
a) Start with form:
d , then u
2
b
2
2

12. Quadratic equations in
in this form:
ax 2 bx c 0, a 0
11.1 #23,25,28,34
Complete the square
# 23 x2
2
1
x2
2
2x
1) The Square Root Property
2
2x
1
2
1
if u 2
2
2
1
d , then u
d
2) Completing the Square
a) Start with form: x 2
bx
b
b) Half b and square it:
2
1
c) Add to x 2
2
bx :
2
It is now a perfect square:
x
2
2x 1
x 1
2
b
x 2 bx
2
d) It is now a perfect square:
x2
bx
b
2
2
x
b
2
2

13. Quadratic equations in
in this form:
ax 2 bx c 0, a 0
11.1 #23,25,28,34
Complete the square
# 25 x
2
14
7
2
1) The Square Root Property
14x
14
1
2
49
if u 2
14
2
7
d , then u
d
2) Completing the Square
a) Start with form: x 2
bx
b
b) Half b and square it:
2
x2 14x 49
c) Add to x 2
2
bx :
2
It is now a perfect square:
x
2
14 x 49
x 7
2
b
x 2 bx
2
d) It is now a perfect square:
x2
bx
b
2
2
x
b
2
2

14. 11.1 #23,25,28,34
Complete the square
# 28 x
2
9x
9
1
2
2
81
9
4
2
x
2
x
9x
81
4
d
a) Start with form: x 2
bx
b
b) Half b and square it:
2
c) Add to x 2
2
bx :
2
b
x 2 bx
2
d) It is now a perfect square:
It is now a perfect square:
2
d , then u
2) Completing the Square
81
4
9x
1) The Square Root Property
if u 2
9
2
9
Quadratic equations in
in this form:
ax 2 bx c 0, a 0
x
9
2
2
x2
bx
b
2
2
x
b
2
2

15. Quadratic equations in
in this form:
ax 2 bx c 0, a 0
11.1 #23,25,28,34
Complete the square
# 34 x
2
9
5
x
2
9
10
9
x
5
1
2
9
x
5
1) The Square Root Property
if u 2
9
10
a) Start with form: x 2
81
100
81
100
bx
b
b) Half b and square it:
2
c) Add to x 2
2
bx :
2
b
x 2 bx
2
d) It is now a perfect square:
It is now a perfect square:
x2
d
2) Completing the Square
2
9
81
x
5
100
d , then u
9
x
10
2
x2
bx
b
2
2
x
b
2
2

16. 3) Solving Quadratic Equations by
Completing the Square
11.1 #35,40,41,44,55
1) The Square Root Property
2
# 35 x 4 x
x2 4x
4
2
x2
2
4x
x2
1
2
x 2
4
36 u 2
2
bx :
2
b
x 2 bx
2
d) It is now a perfect square:
36
36
bx
b
b) Half b and square it:
2
c) Add to x 2
32
d
a) Start with form: x 2
4
2
d , then u
2) Completing the Square
2
4x 4
x 2
if u 2
32
32
4
2
4
Quadratic equations in
in this form:
ax 2 bx c 0, a 0
d ?
then u
d
x2
bx
b
2
2
x
b
2
2

17. 3) Solving Quadratic Equations by
Completing the Square
11.1 #35,40,41,44,55
x 2
x
x
x
1) The Square Root Property
if u 2
36
4
or x
or
d , then u
d
2) Completing the Square
2 6
2 6
Quadratic equations in
in this form:
ax 2 bx c 0, a 0
2 6
x
8
a) Start with form: x 2
bx
b
b) Half b and square it:
2
c) Add to x 2
2
bx :
2
x
8,4
b
x 2 bx
2
d) It is now a perfect square:
x2
bx
b
2
2
x
b
2
2

18. Quadratic equations in
in this form:
ax 2 bx c 0, a 0
3) Solving Quadratic Equations by
Completing the Square
11.1 #35,40,41,44,55
# 40 x
2
1) The Square Root Property
8x 5
x2 8x
8
8 1
2
2
2
16
4
x2 8x 16
x 2 8x 16
2
x 4
x 4
x
4
if u 2
0
d , then u
d
2) Completing the Square
5
a) Start with form: x 2
4
5 16
21
21 u 2 d ?
21 then u
21
bx
b
b) Half b and square it:
2
c) Add to x 2
2
bx :
2
b
x 2 bx
2
d) It is now a perfect square:
d
x2
bx
b
2
2
x
b
2
2

19. Quadratic equations in
in this form:
ax 2 bx c 0, a 0
3) Solving Quadratic Equations by
Completing the Square
11.1 #35,40,41,44,55
# 41 x2
x
2
2x 2
2x
2
2
1
2
1) The Square Root Property
0
2
2
2
1
1
x2 2x 1
x 2 2x 1
2
x 1
x 1
x
1
if u 2
d
2) Completing the Square
a) Start with form: x 2
1
2 1
1
1 u2 d ?
1 then u
i
d , then u
bx
b
b) Half b and square it:
2
c) Add to x 2
2
bx :
2
b
x 2 bx
2
d) It is now a perfect square:
d
x2
bx
b
2
2
x
b
2
2

20. 11.1 #35,40,41,44,55
# 55 8x2 4 x 1 0
8x2 4 x
1
x
x2
x
2
2
1
x
2
1
8
1
1 1
4
2 2
2
1 1
4 16
1
1 1
1
x
2
16
8 16
1
1
1
x
2
16
16
Quadratic equations in
in this form:
ax 2 bx c 0, a 0
1) The Square Root Property
if u 2
d , then u
d
2) Completing the Square
a) Start with form: x 2
bx
b
b) Half b and square it:
2
c) Add to x 2
2
bx :
2
b
x 2 bx
2
d) It is now a perfect square:
x2
bx
b
2
2
x
b
2
2

21. Quadratic equations in
in this form:
ax 2 bx c 0, a 0
11.1 #35,40,41,44,55
# 55 8x2 4 x 1 0
8x2 4 x
1
x
1
4
x
2
2
2
1
x
2
1) The Square Root Property
if u 2
1
8
d
2) Completing the Square
1
16
1
1
x
2
16
2
1
x
4
d , then u
a) Start with form: x 2
1
16
1 2
u
16
bx
b
b) Half b and square it:
2
c) Add to x 2
2
bx :
2
d ?
b
x 2 bx
2
d) It is now a perfect square:
x2
bx
b
2
2
x
b
2
2

22. # 55 8x2
x
x
4x 1 0
1
4
x
2
1
4
1
u2
16
1
then u
16
1
4
Quadratic equations in
in this form:
ax 2 bx c 0, a 0
d ?
1) The Square Root Property
if u 2
d
d , then u
d
2) Completing the Square
a) Start with form: x 2
1
16
1
bx
b
b) Half b and square it:
2
c) Add to x 2
2
bx :
2
x
1
4
1
i
4
b
x 2 bx
2
d) It is now a perfect square:
x2
bx
b
2
2
x
b
2
2

23. •Tonight’s Lecture portion Ch 6rev & 11.1: •DONE
•5 points for Attendance given at:
•8:30 PM
•Homework Assignments due by 8:30PM:•Ch 6rev & 11.1
•Tonight’s Assignments already done? :
•Turn them in BEFORE you leave
•Receive 5 points for attendance
•You may leave anytime AFTER the LECTURE portion
•Arriving late (after 6:45 pm) AND THEN leaving early:
•Receive ZERO points for attendance

24. Quadratic equations in
in this form:
ax 2 bx c 0, a 0
1) The Square Root Property
if u 2
d , then u
d
2) Completing the Square
a) Start with form: x 2
bx
b
b) Half b and square it:
2
c) Add to x 2
bx :
2
2
b
x 2 bx
2
d) It is now a perfect square:
x2
bx
b
2
2
x
b
2
2