Transcript Solving Quadratic Equations by Factoring
Solving Quadratic Equations by Factoring Chapter 9 Lesson 9-5
Introduction This is designed for Mrs. Bonn’s Grade 9 Algebra 1 class. You are expected to follow directions on each slide and navigate through this powerpoint carefully. Your goal is to understand the concept of solving quadratic equations and its relation to the roots (zeros) of quadratic functions and to learn how to solve quadratic equations by factoring.
Objective: • To master the skill of solving quadratic equations by factoring
Required Materials:
• Pencil, Paper, and eraser
The next 6 slides are Warm-Up Review Problems. Being able to successfully solve these problems will ensure that you are ready to work on this lesson.
Lesson Warm-Up #1 Solve this equation for n: 6 + 4
n
= What does n equal?
2
You Are Right!!
6 + 4
n
6 = 2 6 4 4
n n
= 4 4 = 1
Try Again!!
Study this Example: 5 + 3
x
= 5 1 5 3
x
3
x
= 6 3 = 2 6 First, you should subtract 6 from both sides.
+ 4
n
6 = 2 6 4
n
= 4 Now, what should the next step be? What does n equal?
Try Again!!
First, you should subtract 6 from both sides.
6 + 4
n
6 = 2 6 so, 4
n
= 4 Now you need to divide both sides by 4.
What does n equal?
Try Again!!
First, you should subtract 6 from both sides 6 + 4
n
6 = 2 6 4
n
= 4 4 Then, divide both sides by 4 4 What does n equal?
Lesson Warm-Up #2 Solve for a:
a
9 8 What does a equal?
= 4
You Are Right!!!
Solve for a:
a
9 = 8 + 9 +9 4 8 i
a
8 a = 13 i = 104 8
Try Again!!
Study this Example:
y
5 3 = 4 + 3 + 3 NOW: Solve for a:
y
5 i 5
y
= = 35 7 i 5 Step 1: Add 9 on both sides So,
a
8 9 = 4 Think: What is the 2 nd Step you need to do to solve for a?
+ 9 +9
a
8 = 13 What does a equal?
Try Again!
Solve for a: Step 1: Add 9 on both sides Step 2: Multiply 8 to both sides
a
8 + 9 = 4 9 +9 8 i
a
8 = 13 i 8 What does a equal?
Lesson Warm-Up #3 Factor the expression completely: 2
x
2 8 What is the Complete Factored
You Are Right!!!
Factor the expression: 2
x
2 8 2(
x
2 2(
x
4) 2)(
x
+ 2)
Try Again!!!
Factor the expression completely: 2
x
2 Step 1: Factor out the GCF. In this case, GCF = 2. 2(
x
2
x
2 4 Note: Factoring Difference of Perfect 8 4) Squares:
a
2 -
b
2 = (
a
-
b
)(
a
+
b
)
What is the Complete Factored form?
Try Again!!!
Factor the expression completely: 2
x
2 Step 1: Factor out the GCF. In this case, GCF = 2. 2(
x
2 8 4) Step 2: Factor
x
2 4 = (
x
2)(
x
+ 2) So now, what is the Complete Factored form?
Lesson Warm-Up #4 Factor the expression:
x
2 + 4 Is it:
PRIME
OR
You Are Right!!!
Factor the expression:
x
2 + 4 This expression cannot be factored. Therefore, it is
PRIME
Try Again!!!
Notes: Steps to Factor Polynomials: I Un-DISTRIBUTE GCF II Un-FOIL a 2 - b 2 III Un-FOIL ax 2 + bx + c IV Polynomials that cannot be factored are PRIME Hint: Is it possible to factor the given expression? Does it follow any of the factoring rules listed above?
Lesson Warm-Up #5 Factor the expression completely: 2
c
2 + 29
c
+ 14 What is the expression in factored form?
You Are Right!!!
Factor the expression completely: 2
c
2 + 29
c
+ 14 Expression in Factored Form:
Check
: (2
c
+ 1)(
c
+ 14) (2
c
+ 1)(
c
+ 14) 2
c
2 + 28
c
+
c
+ 14 2
c
2 + 29
c
+ 14
Try Again!!
Study the Notes Below: Steps to Factor Polynomials: I Un-DISTRIBUTE GCF II Un-FOIL a 2 - b 2 III Un-FOIL ax 2 + bx + c IV Polynomials that cannot be factored are PRIME Factor completely: 2
c
2
+ 29
c
+ 14
Try Again!!
Factor the expression completely: 2
c
2 + Step 1: (2c+___)(c+___) 29
c
+ 14 You need to think of factors of 14 that will work.
Possibilities: 1 and 14; 2 and 7 Remember to check for the middle term when you FOIL.
Lesson Warm-Up #6 Factor the expression completely: 3
p
2 + 32
p
+ 20 What is the expression in factored form?
You Are Right!!!
Factor the expression: 3
p
2 + 32
p
+ 20
Check
: (3
p
+ 2)(
p
+ 10) 3
p
2 + 30
p
+ 2
p
+ 20 3
p
2 + 32
p
+ 20 (3
p
+ 2)(
p
+ 10)
Try Again!!!
Study the Notes Below: Steps to Factor Polynomials: I Un-DISTRIBUTE GCF II Un-FOIL a 2 - b 2 III Un-FOIL ax 2 + bx + c IV Polynomials that cannot be factored are PRIME Factor completely: 3
p
2
+ 32
p
+ 20
Try Again!!!
Factor the expression completely: 3
p
2 + Step 1: (3p+___)(p+___) 32
p
+ 20 Think: What Factors of 20 will give you the correct middle term?
What is the expression in factored form?
Watch this video to get a sense of what we are going to do for the rest of this lesson: You need to return and continue working on this project after watching the video.
Zero-Product Property For every real number a and b , if ab = 0 , then a = 0 or b = 0 . x + 2 = 0 or x + 3 = 0
Checking Your Understanding #1: If x = 0, then
x
i
y
= __?__
You have 5 seconds to think before the answer appears…
Answer
: 0
Checking Your Understanding #2: If y = 0, then
x
i
y
= __?__
You have 5 seconds to think before the answer appears…
Answer
: 0
What should you do on the “Example” pages: • Study the Example Question carefully and think about how you may want to solve it •The worked out solution(s) and the final answer of the example will appear after a few seconds •Study every step of the solution and the answer carefully so you will be able to solve similar problems later on •When you are ready to continue, navigate to the next slide.
What should you do on the “You Try” pages: • Study the “You Try” Question carefully and think about how you may want to solve it •Solve the problem on the paper that you have with you •The worked out solution(s) and the final answer of the problem will appear after 10-15 seconds •Check your work (step-by-step) with the worked out solution and final answer •If you made mistakes in solving the problem on your paper, you should correct them •When you are ready to continue, navigate to the next slide.
Example 1: Using the Zero-Product Property, Solve: (
x
+ 5)(2
x
6) = 0
x
+
x
5 = 5 = 5
Solutions
: 0
x
5 2
x
6 = 0 + 6 +6 2
x
= 6 = -
x
= 5, 3 3
Example 1 continues: 0
If
Check your Solutions: (
x
+ 5)(2
x
-
x
= 5, then ( 5 + 5)(2 i 5 6) = 0 6) = 0
If
(3
x
+ = 3, then 5)(2 i 3 6) = 0 i 16 = 0 8 i 0 = 0 = 0 0 = 0
Yes
,
x
= 5
Yes
,
x
= 0 3
You Try #1: Using the Zero-Product Property, Solve : (
x
+ 7)(
x
4) = 0
x
+ 7 = 0 or
x
4 =
x
= 7
x
= 0 4
Solution
:
x
= -
7, 4
You Try #2: Using the Zero-Product Property, Solve : (3
y
3
y
5 = 5)(
y
0 or y 2) 2 = = 0 0 + 5 3
y
= + 5 5 y = 2
y
= 5 3
Solution
:
x
= 5 3 , 2
You Try #3: Using the Zero-Product Property, Solve : (6
k
6
k
+ 9 + = 9)(4
k
0 or 4k 11) 11 = 0 = 0 9 6k 9 +11 = 9 4
k
= + 11 = 11
y
= 9 6
k
= 11 4
y
= 3 2
Solution
:
x
= 3 2 , 11 4
Solve :
Quick Check
(
x
5)(
x
2) = 0
Yippee!
YOU GOT IT!
Solve : (
x
5)(
x
2) = 0
x
5 =
x
=
x
= 2, 5 0 or
x
2 = 5
x
= 0 2
Oops!
You need to review the Example and You Try Problems…
Example 2: Solve by Factoring : (
x x
2 8
x
12)(
x
+ 4) 48 = 0 = 0
x
12 = 0 or
x
+ 4
x
= 12
x
= 0 = 4
Solution
:
x
=
12,
-
4
You Try: Solve by Factoring : (
x
-
x
2 + 3)(
x
+
x
12 4) = 0 = 0
x
-
x
= 3 = 0 or
x
+ 3 4
x
= 0 = 4
Solution
:
x
=
3,
-
4
Solve :
Quick Check
x
2 2
x
15 = 0
Yippee!
YOU GOT IT!
Solve : (
x x
2 5)(
x
2
x
+ 3) 15 = 0 = 0
x
5
x
= =
x
= 3, 5 0 or
x
+ 3 = 0 5
x
= 3
Oops!
You need to review the Example and You Try Problems…
Example 3: Solve by Factoring : 2
x
2 Step 1: Subtract 88 from both sides 2
x
2 5
x
88 5
x
= 0 = 88 Step 2: Factor the polynomial Step 3: Set Factors Equal to Zero Step 4: Solve for variable (2 2
x x
+ + 11)( 11 = 11
x
8) = 0 or
x
0 8 11
x
= = 0 8 2
x
= 11
x
11
Solutions
:
x
= 8, 2 = 11 2
You Try: Solve by Factoring :
x
2 12
x
= 36
x
2 12
x
+ (
x
6)(
x
36 6) = = 0 0
x
6
x
= = 0 6
Solution
:
x
= 6
Solve :
Quick Check
x
2 9
x
= 20
Solve :
Yippee!
YOU GOT IT!
x
2
x
2 (
x
9
x
9
x
+ 5)(
x
20 = 20 = 0 4) = 0
x
5 =
x
=
x
= 4, 5 0 or
x
4 = 5
x
= 0 4
Oops!
You need to review the Example and You Try Problems…
Example 4: The diagram below shows a pattern of an open-top box. The total area of the sheet is 288 inches square. The height of the box is 3 in. Therefore, 3-in. by 3-in. squares are cut from each corner. Find the dimensions of the box. Let x = width of a side of the box So, Width of material = x + 3 + 3 = x + 6 Length of material = x + 2 + 3 + 3 Length = x + 8
Example 4 continued: The diagram below shows a pattern of an open-top box. The total area of the sheet is 288 inches square. The height of the box is 3 in. Therefore, 3-in. by 3-in. squares are cut from each corner. Find the (
x
+ 8)(
x
+ 6) = 288
x
2 + 14
x
+ 48 = 288
x
2 + 14
x
(
x
+ 24)(
x
240 10) = = 0 0
x
+
x
24 = 0 or
x
10 = 24 or
x
= 10 = 0
Solution
:
x
= 10, since width ¹
You try: Suppose that a box has a base with a width of x, a length of x + 3, and a height of 1 inch. It is cut from a rectangular sheet of material with an area of 130 inches square. Find the dimensions of the box. Length x Width = Area of Sheet (
x
+ 5)(
x
+ 2) = 130
x
2 + 7
x
+ 10 = 130
x
2 + (
x
7
x
120 8)(
x
+ 15) = = 0 0
x
+ 15
x
= 0 or = 15 or
x
=
x
8 8
Solution
:
x
= = 0 8, since width ¹ negative Dimensions: 8in X 11in X 1in
NOTES: What is the connection between solving quadratic equations by graphing and by factoring?
x
Factoring
2 + 7
x
+ 6 = 0 (
x
+
x
+ 6 6)( =
x
+ 1) = 0 0 or
x
+ 1 = 0
x
= 6
x
= 1
Solution
:
x
= 6, 1
Graphing
y
=
x
2 + 7
x
+ 6
x
int
ercepts
: 1, 6
Check Your Understanding #1: If the roots of the quadratic function g(x) are -2 and 2, what are the solutions of the equation: g(x) = 0
Solutions
:
x
=
2,
-
2
Check Your Understanding #2: If the solution of the quadratic equation h(x) are -3 and -5, what are the zeros of the function: y = h(x)
Zeros
: (
-
3, 0),(
-
5, 0)
Practice Problems #1: Solve : (2
x
+ 3)(
x
4) = 0 2
x
+ 3 = 0 or
x
4 = 0
x
= 3 or
x
2 = 4
Solution
:
x
= 3 , 4 2
Practice Problems #2: Solve by Factoring :
x
2 +
x
42 = 0 (
x
+
x
+ 7 7)( =
x
6) = 0 0 or
x
-
x
= 7 or 6
x
= = 0 6
Solution
:
x
= -
7, 6
Practice Problems #3: Solve by Factoring : 3
x
2 2
x
21 = 0 3
x
2 (3
x
+ 3
x
+ 7 7)(
x
3) = = 0 or
x
0 3 = 0 2
x
= 21
x
= 7 3 or
x
Solution
:
x
= = 3 7 3 , 3
Quiz Time
1. Solve : (2
x
3)(
x
+ 2) = 0
Oops!! Please review the Example below and Try Again!
Using the Zero-Product Property, Solve: (
x
+ 5)(2
x
2
x
6) 6 = = 0 0
x
+ 5 = 0 + 6 +6 Add 6 Subtract 5 5 5
x
= 5 2
x
= 6 Divide By 2
x
= 3
Solutions
:
x
= 5, 3
YES!
YOU ARE RIGHT!
NOW TRY QUESTION 2
Quiz Time 2. Solve by Factoring : 6 =
a
2 5
a
Oops!! Please review the Example below and Try Again!
Solve by Factoring:
x x
2 2 12
x
12
x
+ = 36 36 = 0 (
x
6)(
x
6) =
x
6 = 0 0
x
=
Solution
:
x
6 = 6
YES!
YOU ARE RIGHT!
NOW TRY QUESTION 3
Quiz Time 3. Solve by Factoring : 12
x
+ 4 = 9
x
2
Oops!! Please review the Example below and Try Again!
Solve by Factoring: 2
x
2 Step 1: Subtract 88 from both sides 2
x
2 5
x
5
x
88 = = 0 88 Step 2: Factor the polynomial Step 3: Set Factors Equal to Zero 2 Step 4: Solve for variable (2
x x
+ + 11)( 11 = 11
x
8) = 0 0 or
x
8 11
x
= = 0 8 2
x
= 11
x
= 11 2
YES!
YOU ARE RIGHT!
NOW TRY QUESTION 4
Quiz Time 4. Solve by Factoring : 4
y
2 = 25
Oops!! Please review the Example below and Try Again!
Solve: 3
x
2 = 27 3
x
2 = 27 Divide Both Sides by 3
x
2 = 9 Take the Square Root Both Sides
x
= ± 3
YES!
YOU ARE RIGHT!
You Have Done a Great Job on this StAIR!
Please click Home to return to the First Slide to allow other students to work on this.