1. PREVIEW/SUMMARY OF QUADRATIC EQUATIONS & FUNCTIONS
QUADRATIC – means second power
Recall LINEAR – means first power

2. METHOD 1 - FACTORING Set equal to zero Factor
Use the Zero Product Property to solve (Each factor with a variable in it could be equal to zero.)

3. {0, 3} METHOD 1 - FACTORING 5x2 = 15x 5x2 – 15x = 0 5x (x – 3) = 0
Any # of terms – Look for GCF factoring first!
5x2 = 15x
5x2 – 15x = 0
{0, 3}
5x (x – 3) = 0
5x = 0 OR x – 3 = 0
x = 0 OR x = 3

4. {– 3, 3} METHOD 1 - FACTORING x2 – 9 = 0 (x + 3) (x – 3) = 0
Binomials – Look for Difference of Squares
2. x2 = 9
Conjugates
x2 – 9 = 0
{– 3, 3}
(x + 3) (x – 3) = 0
x + 3 = 0 OR x – 3 = 0
x = – 3 OR x = 3

5. {4 d.r.} METHOD 1 - FACTORING 3. x2 – 8x = – 16 x2 – 8x + 16 = 0
Trinomials – Look for PST (Perfect Square Trinomial)
3. x2 – 8x = – 16
x2 – 8x + 16 = 0
{4 d.r.}
(x – 4) (x – 4) = 0
x – 4 = 0 OR x – 4 = 0
Double Root
x = 4 OR x = 4

6. {-3/2, 0, 5} METHOD 1 - FACTORING 4. 2x3 – 15x = 7x2
Trinomials – Look for Reverse of Foil
4. 2x3 – 15x = 7x2
2x3 – 7x2 – 15x = 0
{-3/2, 0, 5}
(x) (2x2 – 7x – 15) = 0
(x) (2x + 3)(x – 5) = 0
x = 0 OR 2x + 3 = 0 OR x – 5 = 0
x = 0 OR x = – 3/2 OR x = 5

7. METHOD 2 – SQUARE ROOTS OF BOTH SIDES
Reorder terms IF needed
Works whenever form is (glob)2 = c
Take square roots of both sides (Remember you will need a  sign!)
Simplify the square root if needed
Solve for x. (Isolate it.)

8. METHOD 2 – SQUARE ROOTS OF BOTH SIDES
x2 = 9
{-3, 3}
x =  3
Note  means both +3 and -3!
x = -3 OR x = 3

9. METHOD 2 – SQUARE ROOTS OF BOTH SIDES
2. x2 = 18

10. METHOD 2 – SQUARE ROOTS OF BOTH SIDES
3. x2 = – 9
Cannot take a square root of a negative. There are NO real number solutions!

11. METHOD 2 – SQUARE ROOTS OF BOTH SIDES
4. (x-2)2 = 9
{-1, 5}
This means: x = and x = 2 – 3
x = 5 and x = – 1

12. METHOD 2 – SQUARE ROOTS OF BOTH SIDES
Rewrite as (glob)2 = c first if necessary.
5. x2 – 10x + 25 = 9
(x – 5)2 = 9
{2, 8}
x = 8 and x = 2

13. METHOD 2 – SQUARE ROOTS OF BOTH SIDES
Rewrite as (glob)2 = c first if necessary.
6. x2 – 10x + 25 = 48
(x – 5)2 = 48

14. METHOD 3 – COMPLETE THE SQUARE
Goal is to get into the format: (glob)2 = c
Method always works, but is only recommended when a = 1 or all the coefficients are divisible by a
We will practice this method repeatedly and then it will keep getting easier!

15. METHOD 3 – COMPLETE THE SQUARE
Example: 3x2 – 6 = x2 + 12x
Simplify and write in standard form:
ax2 + bx + c = 0
2x2 – 12x – 6 = 0
x2 – 6x – 3 = 0
Set a = 1 by division
Note: in some problems
a will already be equal to 1.

16. METHOD 3 – COMPLETE THE SQUARE
x2 – 6x – 3 = 0
Move constant to other side
Leave space to replace it!
x2 – 6x = 3
x2 – 6x + 9 =
Add (b/2)2 to both sides
This completes a PST!
(x – 3)2 = 12
Rewrite as (glob)2 = c

17. METHOD 3 – COMPLETE THE SQUARE
(x – 3)2 = 12
Take square roots of both sides – don’t forget 
Simplify
Solve for x

18. METHOD 4 – QUADRATIC FORMULA
This is a formula you will need to memorize!
Works to solve all quadratic equations
Rewrite in standard form in order to identify the values of a, b and c.
Plug a, b & c into the formula and simplify!
QUADRATIC FORMULA:

19. METHOD 4 – QUADRATIC FORMULA
Use to solve: 3x2 – 6 = x2 + 12x
Standard Form: 2x2 – 12x – 6 = 0

20. METHOD 4 – QUADRATIC FORMULA

21. REVIEW – QUADRATIC FUNCTIONS
The graph is a parabola. Opens up if a > 0 and down if a < 0.
To find x-intercepts: – may have Zero, One or Two x-intercepts
1. Set y or "f(x)" to zero on one side of the equation
2. Factor & use the Zero Product Prop to find TWO x-intercepts
To find y-intercept, set x = 0. Note f(0) will equal c. I.E. (0, c)
d. To find the coordinates of the vertex (turning pt):
1. x-coordinate of the vertex comes from this formula:
2. plug that x-value into the function to find the y-coordinate
e. The axis of symmetry is the vertical line through vertex: x =

22. REVIEW – QUADRATIC FUNCTIONS
Example Problem: f(x) = x2 – 2x – 8
a. Opens UP since a = 1 (that is, positive)
b. x-intercepts: 0 = x2 – 2x – 8
0 = (x – 4)(x + 2)
(4, 0) and (– 2, 0)
c. y-intercept: f(0) = (0)2 – 2(0) – 8  (0, – 8)
d. vertex:
e. axis of symmetry: x = 1

23. PRACTICE METHOD 2 – SQUARE ROOTS OF BOTH SIDES
x2 = 121
{-11, 11}
x =  11
Note  means both +11 and -11!
x = -11 OR x = 11

24. PRACTICE METHOD 2 – SQUARE ROOTS OF BOTH SIDES
2. x2 = – 81
Cannot take a square root of a negative. There are NO real number solutions!

25. PRACTICE METHOD 2 – SQUARE ROOTS OF BOTH SIDES
Rewrite as (glob)2 = c first if necessary.
3. 6x2 = 156
x2 = 26

26. PRACTICE METHOD 2 – SQUARE ROOTS OF BOTH SIDES

27. PRACTICE METHOD 2 – SQUARE ROOTS OF BOTH SIDES
Rewrite as (glob)2 = c first if necessary.
5. 9(x2 – 14x + 49) = 4
(x – 7)2 = 4/9
{6⅓, 7⅔}

28. PRACTICE METHOD 2 – SQUARE ROOTS OF BOTH SIDES6.

29. PRACTICE METHOD 3 – COMPLETE THE SQUARE
Example: 2b2 = 16b + 6
Simplify and write in standard form:
ax2 + bx + c = 0
2b2 – 16b – 6 = 0
b2 – 8b – 3 = 0
Set a = 1 by division
Note: in some problems
a will already be equal to 1.

30. PRACTICE METHOD 3 – COMPLETE THE SQUARE
b2 – 8b – 3 = 0
Move constant to other side
Leave space to replace it!
b2 – 8b = 3
b2 – 8b + 16 = 3 +16
Add (b/2)2 to both sides
This completes a PST!
(b – 4)2 = 19
Rewrite as (glob)2 = c

31. PRACTICE METHOD 3 – COMPLETE THE SQUARE
(b – 4)2 = 19
Take square roots of both sides – don’t forget 
Simplify
Solve for the variable

32. PRACTICE METHOD 3 – COMPLETE THE SQUARE
Example: 3n2 + 19n + 1 = n - 2
Simplify and write in standard form:
ax2 + bx + c = 0
3n2 + 18n + 3 = 0
n2 + 6n + 1 = 0
Set a = 1 by division
Note: in some problems
a will already be equal to 1.

33. PRACTICE METHOD 3 – COMPLETE THE SQUARE
n2 + 6n + 1 = 0
Move constant to other side
Leave space to replace it!
n2 + 6n = -1
n2 + 6n + 9 =
Add (b/2)2 to both sides
This completes a PST!
(n + 3)2 = 8
Rewrite as (glob)2 = c

34. PRACTICE METHOD 3 – COMPLETE THE SQUARE
(n + 3)2 = 8
Take square roots of both sides – don’t forget 
Simplify
Solve for the variable

35. PRACTICE METHOD 3 – COMPLETE THE SQUARE
What number “completes each square”?
1. x2 – 10x = -3
1. x2 – 10x + 25 =
2. x x = 1
2. x x + 49 =
3. x2 – 1x = 5
3. x2 – 1x + ¼ = 5 + ¼
4. 2x2 – 40x = 4
4. x2 – 20x =

36. PRACTICE METHOD 3 – COMPLETE THE SQUARE
Now rewrite as (glob)2 = c
1. (x – 5)2 = 22
1. x2 – 10x + 25 =
2. x x + 49 =
2. (x + 7)2 = 50
3. x2 – 1x + ¼ = 5 + ¼
3. (x – ½ )2 = 5 ¼
4. x2 – 20x =
4. (x – 10)2 = 102

37. PRACTICE METHOD 3 – COMPLETE THE SQUARE
Show all steps to solve.
⅓k2 = 4k - ⅔
k2 = 12k - 2
k2 - 12k = - 2
k2 - 12k + 36 =
(k - 6)2 = 34

38. PRACTICE METHOD 4 – QUADRATIC FORMULA
Show all steps to solve & simplify.
2x2 = x + 6
2x2 – x – 6 =0

39. PRACTICE METHOD 4 – QUADRATIC FORMULA
Show all steps to solve & simplify.
x2 + x + 5 = 0

40. PRACTICE METHOD 4 – QUADRATIC FORMULA
Show all steps to solve & simplify.
x2 +2x - 4 = 0

41. THE DISCRIMINANT – MAKING PREDICTIONS
b2 – 4ac is called the discriminant
Four cases:
1. b2 – 4ac positive non-square two irrational roots
2. b2 – 4ac positive square two rational roots
3. b2 – 4ac zero one rational double root
4. b2 – 4ac negative no real roots

42. THE DISCRIMINANT – MAKING PREDICTIONS
Use the discriminant to predict how many “roots” each equation will have.
1. x2 – 7x – 2 = 0
49–4(1)(-2)=57 2 irrational roots
2. 0 = 2x2– 3x + 1
9–4(2)(1)=1  2 rational roots
3. 0 = 5x2 – 2x + 3
4–4(5)(3)=-56  no real roots
100–4(1)(25)=0  1 rational double root
4. x2 – 10x + 25=0

43. THE DISCRIMINANT – MAKING PREDICTIONS about Parabolas
The “zeros” of a function are the x-intercepts on it’s graph. Use the discriminant to predict how many x-intercepts each parabola will have and where the vertex is located.
1. y = 2x2 – x - 6
1–4(2)(-6)=49  2 rational zeros
opens up/vertex below x-axis/2 x-intercepts
2. f(x) = 2x2 – x + 6
1–4(2)(6)=-47  no real zeros
opens up/vertex above x-axis/No x-intercepts
3. y = -2x2– 9x + 6
81–4(-2)(6)=129 2 irrational zeros
opens down/vertex above x-axis/2 x-intercepts
4. f(x) = x2 – 6x + 9
36–4(1)(9)=0  one rational zero
opens up/vertex ON the x-axis/1 x-intercept

44. THE DISCRIMINANT – MAKING PREDICTIONS
Note the proper terminology:
The “zeros” of a function are the x-intercepts on it’s graph. Use the discriminant to predict how many x-intercepts each parabola will have.
The “roots” of an equation are the x values that make the expression equal to zero.
Equations have roots. Functions have zeros which are the x-intercepts on it’s graph.

45. FOUR METHODS – HOW DO I CHOOSE?
Some suggestions:
Quadratic Formula – works for all quadratic equations, but first look for a “quicker” method. You must rewrite into standard form before using Quad Formula! Don’t forget to simplify square roots and use value of the discriminant to predict the number of roots or zeros.
Square Roots of Both Sides – use when the problem can easily be put into the form: glob2 = constant.
Examples: 3(x + 2)2=12 or x2 – 75 = 0

46. FOUR METHODS – HOW DO I CHOOSE?
Some suggestions:
Factoring – doesn’t always work, but IF you can see the factors, this is probably the quickest method.
Examples: x2 – 8x = 0 has a GCF
4x2 – 12x + 9 = 0 is a PST
x2 – x – 6 = 0 is easy to FOIL
Complete the Square – It always works, but if you aren’t quick at arithmetic with fractions, then this method is best used when a = 1 and b is even (so no fractions).
Example: x2 – 6x + 1 = 0

47. QUADRATIC FORMULA – Derive it by Completing the Square!
Start with Standard Form:
ax2 + bx + c = 0