Foundations of Math 11
Note Package
Quadratic Functions
Lesson 5 – Solve by Factoring
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

Page 1 – Relating Graphs and Equations
Page 2 – Solve by Factoring
Page 3 – Steps to Solving by Factoring
Homework
o Complete practice questions at end of the lesson with teacher solutions
o Complete all of these practice questions
Lesson 6 – Solve by Completing the Square
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


Page 1 – The Square Root Principle
Page 2 – Solve by Completing the Square
Page 3 – Steps to Solving by Completing the Square
Homework
o Complete practice questions at end of the lesson with teacher solutions
o Complete all of these practice questions
Lesson 7 – The Quadratic Formula




Page 1 – Deriving the Quadratic Formula
Page 2 – Applying the Quadratic Formula
Page 3 – Finding x-Intercepts
Homework
o Complete practice questions at end of the lesson with teacher solutions
o Complete all of these practice questions
Lesson 8 – The Discriminant




Page 1 – Number of Solutions
Page 2 – Applying the Discriminant
Page 3 – Imaginary Numbers
Homework
o Complete practice questions at end of the lesson with teacher solutions
o Complete all of these practice questions
Lesson 5 – Solve by Factoring
1. Write a quadratic function that has zeros:
a.
b.
2. Solve the following quadratic equation:
Solve, find the roots or determine the zeros of the following:
3. Solve. (m-4)(m+2)=0
4. Find the roots of (m+4)(m-7)=0
6. Determine the zeros of
(3m-1)(m-1)=0
9. Solve. x  7 x  8  0
Solution hints:
 Factor.
 Set each factor to zero and
solve.
2
7. Find the x-intercepts of
m(5m-2)=0
10. Determine the roots of
3x 2  11x  4  0
5. Determine the zeros of
(m-3)(m-5)=0
8. Find the roots of
(2m-3)(7m+2)=0
2
11. Solve.  6 x  x  2  0
12. Determine the roots of
2x2 + 11x + 5 = 0
13. Solve. x2 - x - 30 = 0
Write a quadratic equation that has the following solutions.
15. 0 & -2
16. 1 & 3
17. 4 & 4
(m-1)(m-3)=0
14. Find the zeros
4x2 – 5x – 6 = 0
2
&4
18. 3
m 2  4m  3  0
1 1
&
19. 2 2
20. –5 & -1
21. 0 and 0
22. a and b
Lesson 6 – Solve by Completing the Square
1. Solve by completing the square:
2. Solve by completing the square:
3. Solve by completing the square:
Lesson 7
1. Solve using the quadratic formula: 7x2 + 12x + 4 = 0
2. The height of a ball in metres after being thrown by you is given by the function:
f(t) = -4t2 + 9.2t + 1.5 , where t is the time in seconds. How long does it take for the ball to hit the
ground?
Simplify.
12
3.
ThinkWhat is the
biggest perfect
square that goes
into 12?
4.
40
5.
75
6.
27
7.
98
 4 3
2 3
Simplify and solve x.
8.
x
x
5  10 3
5
9.
12 3
1
x 12 3
OR
x 12 3
x 
5  10 3
10
10.
x 
 5  10 3
5
11.
x 
5  25
5
State the x-intercepts of the following (Approximate where necessary):
 
 
2
12. Graph.
2
y  x 3 4
y  x 2 1
13. Graph.
 
2
14. Graph.
y  x 2 6
X-intercepts?
X-intercepts?
X-intercepts?
Given:
Given:
Given:
 
2
 
2
 
2
y  x  3  4  x 2  6x  5
y  x  2  1  x 2  4x  5
y  x  2  6  x 2  4x  2
Find the x-intercepts by factoring
if possible.
Find the x-intercepts by factoring
if possible.
Find the x-intercepts by factoring
if possible.
State the values of a, b, and c. Then use the quadratic formula to solve the equation.
2
16. x  8x  6  0
a=
b=
c=
15. x  6x  5  0
Solution:
Start with a=1, b=6,c=5.
Fill out the Q.F.
2
x
x
6  62  4(1)(5)
2(1)
6  36  20
2(1)
x  1 & 5
 
6  4
2
2
18. 2p  p  45  0
a=
b=
c=
2
17. 6m  7m  2  0
a=
b=
c=
Determine the x-intercepts by graphing and by using the quadratic formula.
2
19. y  2x  12x  16
Turns into
2
20. y  x  6 x  9
Turns into
2
21. y  x  2x  2
Turns into
y  ( x  3) 2
y  (x  1) 2  1
X-intercepts?
X-intercepts?
X-intercepts?
Determine the roots using the QF
Determine the roots using the QF
2x 2  12x  16  0
x  6x  9  0
Determine the roots using
the QF
What do you notice about the final
number under the square root
sign?
What do you notice about
the final number under the
square root sign?
The nature of the roots: Two equal
real roots.
The nature of the roots: No
real roots.
y  2( x  3) 2  2
x
Start
2
b  b  4ac
2
x 2  2x  2  0
2a
What do you notice about the final
number under the square root
sign?
The nature of the roots: Two
different real roots.
Solve.
22. m(6m  7)  2
Possible solution: Expand,
simplify, use QF
2
23. Solve. 6m  7m  5m  2
2
24. Solve. 6m  7m  5m  5
6m 2  7m  2
6m 2  7m  2  0
x
x
x
x
  7  (7)2  4(6)(2)
2(6)
7  49  48
12
7 1
12 
2
3,
x
25. Solve.
x
8
12 or
x
6
12
1
2
(m  3)(5m  1)  (2m  1)(m  7)
26. Solve. (2m  1)(3m  5)  (m  2)(2m  1)
Lesson 8 – The Discriminant
1. What value of k in kx2 - 4x – 2 = 0 would result in...
a. 2 solutions
b. 1 solution
c. no solutions
2. Write the following imaginary numbers in terms of “i”:
 64
a.
8
b.
3. Plotting imaginary numbers and fractals: extra study.
Determine the value of the discriminant and the Nature of the roots.
2
4. Graph. y  2x  12x  16
2
5. Graph. y  x  6x  9
2
6. Graph. y  x  2x  2
y  2x 2  12x  16
y  x 2  6x  9
y  x 2  2x  2
Turns into
Turns into
Turns into
y  2( x  3)  2
2
y  ( x  3)2
y  (x  1) 2  1
State the value of the discriminant.
State the value of the discriminant.
State the value of the discriminant.
State the nature of the roots.
State the nature of the roots.
State the nature of the roots.
Imaginary numbers are undefined in the REAL # system.
i
7.
2
4 
 1 &  1  i
Negatives  imaginary#
8.
 16 =
9.
 25 =
10. 5   25 =
11.
8 =
4   1  2i
12.
 50
13.
 24
14.
1
15. 5   4
16. 5   49
17.
 150
18.
 48
19.
 45
20. 5   28
21. 3   64
Determine the roots of each equation. State whether they are real or imaginary(complex).
2
2
2
22. x  2x  3  0
23. x  2x  1  0
24. x  2x  5  0
Reminder.
25. State the number of
26. State the number of
solutions if b  4ac  0 .
2
28. Write an expression for the
descriminant for
27. State the number of
2
solutions if b  4ac  0 .
solutions if b  4ac  0 .
2
29. Write an expression for the
descriminant for
30. Write an expression for the
descriminant for
kx 2  2x  3  0
2x 2  kx  8  0
Solutions
31. Find the value(s) of K that
leads to 2 different real
2
roots. x  kx  1  0
32. Find the value(s) of K that
leads to 2 equal real roots.
2x 2  kx  k  0
33. Find the value(s) of K that
leads to 2 no real roots.
(2k  1)x 2  8x  6  0
kx2  4 x  k  0
Possible solution strategy:
Possible solution strategy:
Must satisfy b  4ac  0
2
Must satisfy b  4ac  0
2
Must Satisfy b  4ac  0
(k ) 2  4(1)(1)  0
82  4(2k  1)(6)  0
Possible solution strategy:
2
k2 4  0
k2  4
So either k  2
or
k  2
k  2
Two 2 different real roots when
….
k 2
k  2
5…
i.e. 2.1,3,4,5…
i.e. -2.1,-3,-4,-
64  48k  24  0
40  48k  0
40  48k
(4)2  4(k )(k )  0
16  4k 2  0
4k 2  16
4k  16
2
k2  4
So either
So 2 equal real roots when….
k 
40 5

48 6
k 2
or
k  2
k  2
So 0 real roots when…
k  2 or k  2
Find the value(s) of K that leads to the following number of solutions.
34. Find the value(s) of K that
35. Find the value(s) of K that
36. Find the value(s) of K that
leads to 2 different real
leads to 2 equal real roots.
leads to no real roots.
2
roots. x  kx  9  0
37. Find the value(s) of K that
leads to 2 different real
2
roots. kx  4x  3  0
(2k ) x 2  8x  6  0
38. Find the value(s) of K that
leads to 2 equal real roots.
x 2  kx  7  0
kx2  6 x  k  0
39. Find the value(s) of K that
leads to 2 no real roots.
kx 2  8x  9  0
Answers
Lesson 5
1. Teacher Solution
2. Teacher Solution
3. m= 4, -2
4. m=-4, 7
5. m=3, 5
3 2
,
8. m= 2 7
9. x= 8, -1
13. x= 6, -5
3
,2
14. x= 4
1
,1
6. m= 3
7. m=
1
,4
10. x= 3
2 1
,
11. x= 3 2
1
,5
12. x= 2
15. m2+2m=0
16. m2-4m+3=0
17. m2-8m+16=0 18. 3m214m+8=0
20. m2+6m+5=0
21. m2=0
22. m2-(a+b)m+ab=0
0,
2
5
19. 4m2-4m+1=0
Lesson 6
1. Teacher Solution
2. Teacher Solution
3. Teacher Solution
Lesson 7
1. Teacher Solution
2. Teacher Solution
3.
2 3
8.
x 12 3
OR
4.
2 10
9.
X
x 12 3
13. No xintercepts
16.
x  4  22
21. No solution
26.
3 1
 ,
2 2
1
2
 3
14. Decimal xintercepts. See
see the
quadratic
formula to
solve.
17.
2 1
,
3 2
22. x 
2
3
, x
1
2
5.
5 3
10.
X  1  2 3
6. 3 3
11. 0,2
7. 7 2
12. -5 & -1
19. 2,4,
20. –3,-3
15. –1, -5
18.
9
 ,5
2
23.
3 6 3 6
,
3
3
24.
6  66 6  66
,
6
6
25.
1, 
4
3
Lesson 8
1. Teacher Solution
2. Teacher Solution
3. Teacher Solution
4. 16, 2 different
real roots.
8. 4i
13.
18.
2i 6
4i 3
5. 0, 2 equal real
roots.
6. –4, No real
roots.
7. i
9. 5i
10. 5+5i
11.
2i 2
12.
14. i
15. 5+2i
16. 5+7i
17.
19.
3i 5
23. 1 (double
root), REAL
24. 1+2i, 1-2i,
COMPLEX
2
28. k  64
29. 4  12k
20.
5  2i 7
21.
3  8i
25. 2equal
26. 2dif
2
30. k  8k
31. k>2, k<-2
38.
34. k>6 or k<-6
k  2 7
39.
k 
16
9
4
35. k= 3
36. -3<k<3
5i 6
22. 3, -1. REAL
27. No real
solutions

32.
33. 2  k  2
5i 2
37.
5
6
k 
4
3