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Quadratic Equations
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Quadratic EquationsTable of Contents
Learning Objectives
Finding a, b and c
Finding the vertex and min/max values
Finding the discriminant
Finding the roots
Factoring
Completing the Square
Quadratic Formula
Quadratics with Function Tables
Graphing Quadratics
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Learning Objectives
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Learning Objectives
LO 1
LO 2
Understand what a Quadratic Equation represents
Perform basic operations with Quadratic Equations
LO 3 Build a Quadratic equation using various techniques
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Definitions
Definition 1 Quadratic Equations are in the form ax² + bx + c = 0
TOC
Definition 2 “a” determines the direction and Magnitude (width) of the curve
Definition 3 “-b/2a” determines whether the min/max has ± x value : note the OPPOSITE of b!
Definition 4 “c” determines whether the y intercept has a ± y value
Definition 5 The Discriminant determines the number of roots (b² - 4ac)
Definition 6 The Roots (x intercepts) are determined by factoring the equation
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Previous knowledge
PK 1 Basic Operations and Properties
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Rule 1
Rule 2
Know how to find a, b, -b, and c
Know how to find the vertex (-b/2a) and the discriminant (b² - 4ac)
Rule 3 Know how to find the roots by factoring, completing the square and the formula
Basic Rules of Quadratics
TOC
Rule 4 Always check your work
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Quadratic Equations
• Quadratic Equations f(x) = ax² + bx + c
– You are given certain information in a function f(x)
• Width of the curve is determined by (a)
• Symmetry is determined by -b/2a (note the opposite of b)
• Y intercept of the curve is determined by (c)
• Remember all function tables are the same regardless of the equation
• Go up or down the same amount and look for a pattern
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Finding a, b, -b and c
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Q1
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QuadraticsFinding a, b and c
In the quadratic equation below find a, b, -b and c
f(x) = 3x² + 5x + 2
a =
b =
-b =
c =
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Q1 Answer
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QuadraticsFinding a, b and c
In the quadratic equation below find a, b and c
f(x) = 3x² + 5x + 2
a =
b =
-b =
c =
3 25
TOC
- 5
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Now you try
f(x) = -2x² - 4x - 8
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QuadraticsFinding a, b and c
In the quadratic equation below find a, b and c
f(x) = -2x² -4x -8
a =
b =
-b =
c =
-2 -8-4
TOC
4
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Now you try
f(x) = -¾ x² - ¼x + ½
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QuadraticsFinding a, b and c
In the quadratic equation below find a, b and c
f(x) = -¾ x² - ¼x + ½
a =
b =
-b =
c =
-¾ + ½- ¼¼
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Changes in a
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0,0
f(x) = 4x²f(x) = - 4x²
Step 1 – Compare Graphs
Graph 4x²
Graph - 4x²
Step 2 - Compare a values
a = 4
a = - 4
Step 3 – Conclusion
When a is + , curve goes up
When a is - , curve goes down
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0,0
f(x) = 4x²f(x) = - 1/4x²
Step 1 – Compare Graphs
Graph 4x²
Graph - 1/4x²
Step 2 - Compare a values
a = 4
a = - 1/4
Step 3 – Conclusion
When a is large, curve narrows
When a is small, curve widens
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Changes in b
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0,0
f(x) = x² + 2xf(x) = x² - 4x
Step 1 – Compare Graphs
Graph x² + 2x
Graph x² - 4x
Step 2 - Compare b values
b =+2b= - 4
Step 3 – Question
If b is + , why is the curve on the negative side of x axis?
If b is -, why is the curve on the positive side of x axis?
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0,0
f(x) = x² - 2xf(x) = x² + 4x
Step 1 – Compare Graphs
Previous Graph x² + 2x
Previous Graph x² - 4x
Step 2 - Compare b values
b = -2
b = + 4
Step 3 – Conclusion
If b is + , why is the curve on the negative side of x axis?
If b is -, why is the curve on the positive side of x axis?
Hint: use –b/2aTOC
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Changes in c
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0,0
f(x) = x² + 2xf(x) = x² - 2x - 1
Step 1 – Compare Graphs
Previous Graph x² + 2x
New Graph x² - 2x - 1
Step 2 - Compare c values
c = 0
c = - 1
Step 3 – Conclusion
C represents where the curve crosses the y axis
+ c means positive y intercept
No c means the origin
- c means negative y intercept
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Q2
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0,0
f(x) = x² + 2x - 3
Step 1 – Is this function:
Up or down?
Wide, Normal or Narrow?
Left or right of 0,0?
Positive or negative y intercept?
Step 2 - Answers:
Up
Normal
Left of 0,0
Y intercept - 3
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Q 3
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0,0
f(x) = ?
Step 1 – What are the values:
a is/is not a fraction
a <0 , a = 0 , a>0
b < 0, b = 0, b>0
C <0, c = 0, c>0
Step 2 - Answers:
a is a fraction
a<0 (meaning a is negative)
b = 0
c >0
The equation is:
f(x) = - 1/4x² + 5TOC
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Finding the Vertex
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Q4
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0,0
f(x) = x² + 2x - 3
What is the vertex of the equation?
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Q4 Answer
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0,0
f(x) = x² + 2x - 3
What is the vertex of the equation?
TOC
-1, -4
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0,0
f(x) = x² + 2x - 3
Step 1 – The vertex
The vertex is a coordinate value
Minimums go with + a values
Maximums go with – a values
a in this equation is +
Step 2 – The formula
Identify -b in the equation
Use the formula (-b/2a)
This is the x value (-1 )
Plug x into equation; find y
f(x) = x² + 2x – 3
f(x) = (-1)² + 2(-1) -3 = -4
This is the y value
TOC
min
- 2
-2 2(1)
-1,
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Now you try
x² + 6x + 8
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0,0
f(x) = x² + 6x + 8
Step 1 – The vertex
Minimums go with + a values
a in this equation is +
Step 2 – The formula
Identify -b in the equation
Use the formula(-b/2a)
This is the x value (-3 )
Plug x into equation; find y
f(x) = x² + 6x + 8
f(x) = (-3)² + 6(-3) + 8 = -1
This is the y value TOC
min
- 6
-6 2(1)
-3,
-1
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Now you try
- 2x² - 8x - 4
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0,0
f(x) = - 2x² - 8x - 4
Step 1 – The vertex
Maximums go with - a values
a in this equation is -
Step 2 – The formula
Identify -b in the equation
Use the formula (-b/2a)
This is the x value (-2 )
Plug x into equation; find y
f(x) = - 2x² - 8x - 4
f(x) = -2(-2)² - 8(-2) - 4 = 4
This is the y value TOC
max
+ 8
8 2(-2)
-2,
4
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Last one
x² - 4
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0,0
f(x) = x² - 4
Step 1 – The vertex
Minimums go with + a values
a in this equation is +
Step 2 – The formula
Identify -b in the equation
Use the formula (-b/2a)
This is the x value (0 )
Plug x into equation; find y
f(x) = x² - 4
f(x) = (0)² - 4 = - 4
This is the y value
No middle term means?
The vertex is the y intercept!
TOC
min
+ 0
0 2(1)
0,
- 4
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Q5
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0,0
f(x) = x² + 2x - 3
What is the vertex of the equation?
Is it a min or max point?
Answers:
Vertex = (-1, -4)
Minimum
TOC
min
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Q6
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0,0
f(x) = - 2x² - 8x - 4
TOC
maxWhat is the vertex of the equation?
Is it a min or max point?
Answers:
Vertex = (-2, 4)
Maximum
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Finding the Discriminant
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Q7
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0,0
f(x) = x² + 2x - 3
What is the discriminant?
How many roots are there?
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Q7 Answer
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0,0
f(x) = x² + 2x - 3
TOC
What is the discriminant?
How many roots are there?
Answers:
discriminant is +
There are two roots
-3, 0 1, 0
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0,0
f(x) = - 2x² - 8x - 4
Step 1 – The discriminant
Discriminant give # of roots
+ means 2 roots
0 means 1 root
- means no roots
Step 2 – The formula b² - 4ac
Locate a, b, -b and c
a = -2b = -8-b = 8C = -4
Substitute into the formula
(-8)² - 4(-2)(- 4)
64 – 32 = + means two roots
Roots are x intercepts
Watch your signs!!!!
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Now you try
x² - 4
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0,0
f(x) = x² - 4
TOC
Step 1 – The discriminant
Discriminant give # of roots
+ means 2 roots
0 means 1 root
- means no roots
Step 2 – The formula b² - 4ac
Locate a, b, -b and c
a = 1b = 0-b = 0C = -4
Substitute into the formula
(0)² - 4(1)(- 4)
0 + 16 = + means two roots
Roots are x intercepts
Watch your signs!!!!
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Now you try
x² + 4x + 4
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0,0
f(x) = x² + 4x + 4
TOC
Step 1 – The discriminant
Discriminant give # of roots
+ means 2 roots
0 means 1 root
- means no roots
Step 2 – The formula b² - 4ac
Locate a, b, -b and c
a = 1b = 4-b = -4C = 4
Substitute into the formula
(4)² - 4(1)(4)
16 - 16 = 0 means one root
One root touches the x axis
Did you recognize the equation?
This is a perfect square (x + 2)²lntaylor © 03/03/12
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Last one
- x² + x - 1
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0,0
f(x) = -x² + x - 1
TOC
Step 1 – The discriminant
Discriminant give # of roots
+ means 2 roots
0 means 1 root
- means no roots
Step 2 – The formula b² - 4ac
Locate a, b, -b and c
a = - 1b = 1-b = -1C = -1
Substitute into the formula
(1)² - 4(-1)(-1)
1 - 4 = - means no roots
No roots means no x intercepts
Did you watch your signs?
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Q8
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0,0
f(x) = x² - 6x + 9
What is the discriminant?
How many roots are there?
Answers:
discriminant is 0
One root
TOC
3, 0
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Q9
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0,0
f(x) = - 2x² - 8x - 4
TOC
What is the discriminant?
How many roots?
Answers:
Discriminant is +
Two roots
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Finding the Roots
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Roots of a Quadratic Equation
Definition
Factor
Roots are x intercepts
Factoring is the quickest way to finding the roots
CTS Completing the Square is easy – once you get the hang of it
TOC
QF Quadratic Formula is the most versatile way of finding the roots
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Factoring Quadratics
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x2 + 6x + 8
Step 1 Check to see if last term is positive
+
Step 2 Divide middle term coefficient by 2
+ 6 2 = 3
Step 3 Square the answer and check last term
(3)(3) = 9
8
Step 4 If they are both the same you have your factors
No match
Step 5 If they are not the same subtract one and add one
(2)(4) = 8
Step 6 Continue until numbers match
Yes
Step 7 Add an x to each parenthesis
(x + 2)(x + 4)
Step 8 Set each () = 0 and solve
x + 2 = 0 and x + 4 = 0 x = - 2 and x = - 4
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Now you try
x2 – 20x + 96
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x2 – 20x + 96
Step 1 Check to see if last term is positive
+
Step 2 Divide middle term coefficient by 2
– 20 2 = – 10
Step 3 Square the answer and check last term
(-10)(-10) = 100
96
Step 4 If they are both the same you have your factors
Step 5 If they are not the same subtract one and add one
(-11)(-9) = 99
Step 6 Continue until numbers match
Yes
(-12)(-8) = 96
Step 7 Add an x to each parenthesis
(x – 12)(x – 8)
Step 8 Set each () = 0 and solve
x - 12 = 0 and x - 8 = 0 x = 12 and x = 8
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Now you try
x2 + 15x + 56
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x2 + 15x + 56
Step 1 Check to see if last term is positive
+
Step 2 Divide middle term coefficient by 2
+ 15 2 = 7.5
Step 3 Since the answer includes 0.5 round up and down
(8)(7) = 56
56
Step 4 If they are both the same you have your factors
Step 5 If they are not the same subtract one and add one
Step 6 Continue until numbers match
Yes
Step 7 Add an x to each parenthesis
(x + 8)(x + 7)
Step 8 Set each () = 0 and solve
x + 8 = 0 and x + 7 = 0 x = - 8 and x = - 7
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Now you try
x2 – 19x + 84
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x2 – 19x + 84
Step 1 Check to see if last term is positive
+
Step 2 Divide middle term coefficient by 2
– 19 2 = – 9.5
Step 3 Since the answer includes 0.5 round up and down
(-10)(-9) = 90
84
Step 4 If they are both the same you have your factors
Step 5 If they are not the same subtract one and add one
(-11)(-8) = 88
Step 6 Continue until numbers match
Yes
(-12)(-7) = 84
Step 7 Add an x to each parenthesis
(x – 12)(x – 7)
Step 8 Set each () = 0 and solve
x - 12 = 0 and x - 7 = 0 x = 12 and x = 7
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Factoring Negative Constants
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x2 – 2x – 48
Step 1 Check to see if last term is negative
–
Step 2 Construct a table for the “difference” of factors for the constant
2 24 22
3 16 13
4 12 8
6 8 2
Step 3 Match the middle term coefficient to the last column
2
Step 4 Use the numbers in the 1st and 2nd columns; add a variable to each
(x 6)(x 8)
Step 5 Put the middle term sign next to the largest number
–
Step 6 Put the opposite sign next to the smallest number!!!!
+
Step 7 Set each () = 0 and solve
(x + 6) = 0 and (x – 8) = 0 x = -6 and x = 8
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Now you try
x2 – 11x – 12
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x2 – 11x – 12
Step 1 Check to see if last term is negative
–
Step 2 Construct a table for the “difference” of factors for the constant
1 12 11
2 6 4
3 4 1
Step 3 Match the middle term coefficient to the last column
11
Step 4 Use the numbers in the 1st and 2nd columns; add a variable to each
(x 1)(x 12)
Step 5 Put the middle term sign next to the largest number
–
Step 6 Put the opposite sign next to the smallest number!!!!
+
Step 7 Set each () = 0 and solve
(x + 1) = 0 and (x - 12) = 0 x = - 1 and x = 12
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Special cases
x2 – 49
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x2 – 49
Step 1 There are only two terms and the last term is negative
Step 2 Square root the last term
49½
= ± 7
Step 3 Write one factor with + and one factor with –
(x + 7)(x – 7)
Step 4 Set each () = 0 and solve
(x + 7) = 0 and (x – 7) = 0 x = - 7 and x = 7
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Now you try
x2 – 225
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x2 – 225
Step 1 There are only two terms and the last term is negative
Step 2 Square root the last term
225½
= ± 15
Step 3 Write one factor with + and one factor with –
(x + 15)(x – 15)
Step 4 Set each () = 0 and solve
(x + 15) = 0 and (x – 15) = 0 x = - 15 and x = 15
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Completing the Square
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Completing the Square
Step 1 Move the constant
Step 2 Take b/2; square the answer and add to both sides
Step 3 Factor out the perfect square; square root and solve
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x² + 6x
Step 1 Set the equation = 0
Step 2 Move c to the opposite side
Step 3 Divide the middle term coefficient by 2
62
= + 3
x² + 6x
+ 5
(x+3)² = 4
x + 3 = 2 and x + 3 = -2
x = -1 and x = - 5
lntaylor © 03/03/12
=
- 5
0
Step 4 Square the number and add to both sides
² = + 9
+ 5
= - 5 + 9
Step 5 Factor the perfect square
Step 6 Square root and solve
+ 9
TOC
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Now you try!
f(x) = x² + 2x - 15
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x² + 2x
Step 1 Set the equation = 0
Step 2 Move c to the opposite side
Step 3 Divide the middle term coefficient by 2
22
= + 1
x² + 2x
- 15
(x+1)² = 16
x + 1 = 4 and x + 1 = - 4
x = 3 and x = - 5
lntaylor © 03/03/12
=
15
0
Step 4 Square the number and add to both sides
² = + 1
- 15
= 15 + 1
Step 5 Factor the perfect square
Step 6 Square root and solve
+ 1
TOC
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Last one!
f(x) = x² -10x + 18
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x² - 10x
Step 1 Set the equation = 0
Step 2 Move c to the opposite side
Step 3 Divide the middle term coefficient by 2
- 10 2
= - 5
x² - 10x
+ 18
(x - 5)² = 7
x - 5 = + √ 7 and x - 5 = - √ 7
x = 5 + √ 7 and x = 5 - √7
lntaylor © 03/03/12
=
- 18
0
Step 4 Square the number and add to both sides
² = + 25
+ 18
= - 18 + 25
Step 5 Factor the perfect square
Step 6 Square root and solve
+ 25
TOC
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Quadratics
The Quadratic Formula _______-b ± √b² - 4ac
2a
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Quadratic Formula
Step 1 Train yourself to write down a, b, -b, and c
Step 2 Set up the equation slowly; make sure signs are correct
Step 3 Set up and Solve Slowly!!!! Make sure the signs are correct!!!!
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x² + 2x - 15
Step 1 Write down a, b, -b and c
Step 2 Write down the formula
Step 3 Set up the equation to solve
Watch your signs!!!!
a = 1b = 2-b = -2c = -15
______-b ± √b² - 4ac 2a
__________-2 ± √2² - 4(1)(-15) 2(1)
Step 4 Solve SLOWLY
Watch your signs!!!! (wait for the program)
_____-2 ± √4 + 60 2
__-2 ± √64 2
-2 ± 8 2
-2 + 8 2
-2 - 8 2
6 2
- 10 2
x = 3 x = - 5
The curve crosses the x axis at 3,0 and -5,0
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Now you try
f(x) = x² + 6x + 9
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x² + 6x + 9
Step 1 Write down a, b, -b and c
Step 2 Write down the formula
Step 3 Set up the equation to solve
Watch your signs!!!!
a = 1b = 6-b = -6c = 9
______-b ± √b² - 4ac 2a
__________-6 ± √6² - 4(1)(9) 2(1)
Step 4 Solve SLOWLY
Watch your signs!!!! (wait for the program)
_____-6 ± √36 - 36 2
__-6± √ 0 2
-6 ± 0 2
-6 2
x = - 3
The curve touches the x axis at -3,0
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Last one!
f(x) = -2x² + 3x - 15
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-2x² - 3x - 15
Step 1 Write down a, b, -b and c
Step 2 Write down the formula
Step 3 Set up the equation to solve
Watch your signs!!!!
a = -2b = -3-b = 3c = -15
______-b ± √b² - 4ac 2a
____________ 3 ± √(-3)² - 4(-2)(-15) 2(-2)
Step 4 Solve SLOWLY
Watch your signs!!!! (wait for the program)
_____ 3 ± √9 - 120 2
____-2 ± √ -111 2
Cannot square root a negative!
The curve does not cross the x axis
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Build a Function Table
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f(x) = x² - 5x - 6
Step 1 – Construct Table
Build 3 column Table
Build Headings
Step 2 – Choosing x values
Start with x = 0
Plug 0 into equation
Solve for y
Build x column
Build middle column
Build y column
Check your work!
Note that every x has one y
x and y together are called ordered pairs (coordinates and a single point on a graph)
TOC
x² - 5x - 6
x f(x) or y
0 0² -5(0) -6 -6
-3
-2
-1
0
1
2
3
(-3)² -5(-3) - 6
(-2)² -5(-2) - 6
(-1)² -5(-1) - 6
(0)² -5(0) - 6
(1)² -5(1) - 6
(2)² -5(2) - 6
(3)² -5(3) - 6
18
8
0
-6
-10
-12
-13
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Now you try!
f(x) = x² + 2x + 1
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f(x) = x² + 2x + 1
Step 1 – Construct Table
Build 3 column Table
Build Headings
Step 2 – Choosing x values
Start with x = 0
Plug 0 into equation
Solve for y
Build x column
Build middle column
Build y column
Check your work!
Note that every x has one y
x and y together are called ordered pairs (coordinates and a single point on a graph)
TOC
x² + 2x + 1
x f(x) or y
0 0² + 2(0) +1 1
-3
-2
-1
0
1
2
3
(-3)² + 2(-3) + 1
(-2)² + 2(-2) + 1
(-1)² + 2(-1) + 1
(0)² + 2(0) + 1
(1)² + 2(1) + 1
(2)² + 2(2) + 1
(3)² + 2(3) + 1
4
1
0
1
4
9
16
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Now you try!
f(x) = x² - 4
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f(x) = x² - 4
Step 1 – Construct Table
Build 3 column Table
Build Headings
Step 2 – Choosing x values
Start with x = 0
Plug 0 into equation
Solve for y
Build x column
Build middle column
Build y column
Check your work!
Note that every x has one y
x and y together are called ordered pairs (coordinates and a single point on a graph)
TOC
x² - 4
x f(x) or y
0 0² - 4 - 4
-3
-2
-1
0
1
2
3
(-3)² - 4
(-2)² - 4
(-1)² - 4
(0)² - 4
(1)² - 4
(2)² - 4
(3)² - 4
5
0
- 3
- 4
- 3
0
5
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Graphing a Quadratic
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0,0
f(x) = x² + 2x - 3
Step 1 – Graph
Write down:
a = 1
b = 2
-b = -2
c = -3
Step 2 – Key measures:
-b/2a = -2/2(1) = -1
Plug in x to find y = -4
Min = (-1, -4)
b² - 4ac = + two roots
Factor (x+3) (x-1)
Solve x = -3 and x = 1
Y intercept - 3 TOClntaylor © 03/03/12
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Now you try!
f(x) = x² - 4
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0,0
f(x) = x² - 4
Step 1 – Graph
Write down:
a = 1
b = 0
-b = 0
c = -4
Step 2 – Key measures:
-b/2a = 0/2(1) = 0
Plug in x to find y = -4
Min = (0, -4)
b² - 4ac = + two roots
Factor (x+2) (x-2)
Solve x = -2 and x = 2
Y intercept - 4 TOClntaylor © 03/03/12
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Now you try!
f(x) = - x² + x - 1
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0,0
f(x) = -x² + x - 1
TOC
Step 1 – Graph
Write down:
a = - 1
b = 1
-b = - 1
c = - 1
Step 2 – Key measures:
-b/2a = - 1/2(- 1) = 1/2
Plug in x to find y = - 3/4
Min = (1/2, - 3/4)
b² - 4ac = - no roots
Pick another x value and solve
x = 2; y = -3 (2, -3)
Y intercept - 1
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Post Test
TOClntaylor © 03/03/12