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Concept Category 4
Quadratic
Equations
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§ 1
Solving Quadratic
Equations by the Square
Root Property
![Page 3: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/3.jpg)
Martin-Gay, Developmental Mathematics 3
Square Root Property
We previously have used factoring to solve
quadratic equations.
This chapter will introduce additional
methods for solving quadratic equations.
Square Root Property
If b is a real number and a2 = b, then
ba
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Martin-Gay, Developmental Mathematics 4
Solve x2 = 49
2x
Solve (y – 3)2 = 4
Solve 2x2 = 4
x2 = 2
749 x
y = 3 2
y = 1 or 5
243 y
Square Root Property
Example
![Page 5: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/5.jpg)
Martin-Gay, Developmental Mathematics 5
Solve x2 + 4 = 0
x2 = 4
There is no real solution because the square root
of 4 is not a real number.
Square Root Property
Example
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Martin-Gay, Developmental Mathematics 6
Solve (x + 2)2 = 25
x = 2 ± 5
x = 2 + 5 or x = 2 – 5
x = 3 or x = 7
5252 x
Square Root Property
Example
![Page 7: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/7.jpg)
Martin-Gay, Developmental Mathematics 7
Solve (3x – 17)2 = 28
72173 x
3
7217 x
7228 3x – 17 =
Square Root Property
Example
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§ 2
Solving Quadratic
Equations by the
Quadratic Formula
![Page 9: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/9.jpg)
Martin-Gay, Developmental Mathematics 9
The Quadratic Formula
Another technique for solving quadratic
equations is to use the quadratic formula.
The formula is derived from completing the
square of a general quadratic equation.
![Page 10: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/10.jpg)
Martin-Gay, Developmental Mathematics 10
A quadratic equation written in standard
form, ax2 + bx + c = 0, has the solutions.
a
acbbx
2
42
The Quadratic Formula
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Martin-Gay, Developmental Mathematics 11
Solve 11n2 – 9n = 1 by the quadratic formula.
11n2 – 9n – 1 = 0 set one side = 0
a = 11, b = -9, c = -1
)11(2
)1)(11(4)9(9 2
n
22
44819
22
1259
The Quadratic Formula
Example
![Page 12: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/12.jpg)
Martin-Gay, Developmental Mathematics 12
Two kinds of answers:
Decimal Answers (to graph):
Simplified Radical Answers (SAT, ACT,
and other college placement exams):
9 125 9 11.2 9 11.2 9 11.2
22 22 22 22
0.9 0.1
and
and
9 5 25 9 5 5
22 22
![Page 13: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/13.jpg)
Martin-Gay, Developmental Mathematics 13
Practice: Solve x using QFormula
Present your answers in
Decimals and Simplified Radicals:
2
2
] ( ) 4 12 63
b] 2 12 46
a f x x x
y x x
![Page 14: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/14.jpg)
Martin-Gay, Developmental Mathematics 14
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Martin-Gay, Developmental Mathematics 15
1/18 Practice Now:
Vertex point?
x-intercept points?
y-intercept point?
2 2] ( ) 10 21 ] ( ) 2( 3) 5a f x x x b g x x
*Vertex point?
*x-intercept points?
*y-intercept point?
![Page 16: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/16.jpg)
Martin-Gay, Developmental Mathematics 16
)1(2
)20)(1(4)8(8 2
x
2
80648
2
1448
2
128 20 4 or , 10 or 2
2 2
x2 + 8x – 20 = 0 multiply both sides by 8
a = 1, b = 8, c = 20
8
1
2
5Solve x2 + x – = 0 by the quadratic formula.
Quadratic Formula – SAT example
Example
![Page 17: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/17.jpg)
Martin-Gay, Developmental Mathematics 17
Concept Category 4 Quadratics
Standard Form of Quadratic Equation
Vertex Form of Quadratic Equation
Vertex: transformation first
Radical Operations
Solving Radical Equations
Nth Roots
Complex Numbers
int : Factoring or Quad: ratic Formula2
bVertex x
a
![Page 18: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/18.jpg)
Martin-Gay, Developmental Mathematics 18
4
16
25
100
144
= 2
= 4
= 5
= 10
= 12
Radicals
![Page 19: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/19.jpg)
Martin-Gay, Developmental Mathematics 19
Perfect Squares
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
324
400
625
289
![Page 20: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/20.jpg)
Martin-Gay, Developmental Mathematics 20
8
20
32
75
40
=
=
=
=
=
2*4
5*4
2*16
3*25
10*4
=
=
=
=
=
22
52
24
35
102
Perfect Square Factor * Other Factor
LE
AV
E I
N R
AD
ICA
L F
OR
M
![Page 21: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/21.jpg)
Martin-Gay, Developmental Mathematics 21
Simplify each expression
737576 78
62747365 7763
A]
B]
![Page 22: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/22.jpg)
Martin-Gay, Developmental Mathematics 22
+ To combine radicals: combine the
coefficients of like radicals
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Martin-Gay, Developmental Mathematics 23
Simplify each radical first, then combine.
323502 2 25* 2 3 16 * 2
2 *5 2 3* 4 2
10 2 12 2
2 2
![Page 24: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/24.jpg)
Martin-Gay, Developmental Mathematics 24
Practice NOW
485273 3 9 * 3 5 16 * 3
3* 3 3 5* 4 3
9 3 20 3
29 3
![Page 25: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/25.jpg)
Martin-Gay, Developmental Mathematics 25
* Multiply the coefficients and then
multiply the radicands and then simplify
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Martin-Gay, Developmental Mathematics 26
35*5 175 7*25 75
Multiply and then simplify
73*82 566 14*46
142*6 1412
204*52 8 100 8*10 80
![Page 27: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/27.jpg)
Martin-Gay, Developmental Mathematics 27
2X
6Y
264 YXP
244 YX
10825 DC
= X
= Y3
= P2X3Y
= 2X2Y
= 5C4D5
![Page 28: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/28.jpg)
Martin-Gay, Developmental Mathematics 28
3X
XX
=
=
XX *2
YY 45Y
=
= YY 2
![Page 29: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/29.jpg)
Martin-Gay, Developmental Mathematics 29
33YPX
2712 YX
9825 DC
=
=
= 5Y
PXYYX *22
5Y
PXYXY=
![Page 30: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/30.jpg)
Martin-Gay, Developmental Mathematics 30
Happy Wed. 1/25
Did everyone have a chance to work on
yesterday’s SMC practice test?
Placement Exams for English and Math are
required for ALL CA colleges (2 or 4 yr)
This practice test have other parts: 2nd part of
Algebra, Geometry, and Precalculus
The less you score and more courses ($$$$ +
Time) you will need to make up for
![Page 31: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/31.jpg)
Martin-Gay, Developmental Mathematics 31
CA College Placement Exams v.s. CAASP
Why they all have similar questions?
Because your HIGH SCHOOL
standards/State Exams are mostly
set by public colleges
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Martin-Gay, Developmental Mathematics 32
2
5 5*5 25 5
4
3 7 3 7 *3 7 *3 7 *3 7 81*49
3969
2
48w 4 48 * 8w w 864w
48w
2
2 3x 2 3 *2 3x x 24 9x 12x
![Page 33: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/33.jpg)
Martin-Gay, Developmental Mathematics 33
2 2
3 5 4 20 3 45 4 3
Practice NOW
4
43 2 2 16x x
![Page 34: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/34.jpg)
Martin-Gay, Developmental Mathematics 34
SOLUTIONS
2
3 5
316x
![Page 35: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/35.jpg)
Martin-Gay, Developmental Mathematics 35
A] Divide the
coefficients, divide the
radicands if possible,
B] rationalize the
denominator so that the
denominator is always
an integer!
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Martin-Gay, Developmental Mathematics 36
Fractional Radicands
22
3
2
2
2 4
3 9 Simplify exponents/radicals
first. Then reduce.
4
9
4 2
39
20
36 20
36
4 5 2 5 5
6 6 3
![Page 37: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/37.jpg)
Martin-Gay, Developmental Mathematics 37
7
568 2*4 22
![Page 38: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/38.jpg)
Martin-Gay, Developmental Mathematics 38
15 15
749
You Try:
63
812
45
16
15
49
45 9 5 3 5
416 16
2 63 2 9 7 2 3 7 6 7 2 7
9 9 9 381
![Page 39: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/39.jpg)
Martin-Gay, Developmental Mathematics 39
How about….
50
3
5 2
3
When simplifying radicals, there can never be a radical in the denominator of the final answer.
BIG
NO NO!
Rationalization: the steps to change a radical denominator to a whole number.
![Page 40: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/40.jpg)
Martin-Gay, Developmental Mathematics 40
24 4 * 6 2 6
28 4 * 7 2 7
Method:
Multiply the
Denominator
by the same
radical
7
6
7*
7
42
49
7
42
In the world of SAT, ACT, math placement exams:
Fraction
expansion
![Page 41: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/41.jpg)
Martin-Gay, Developmental Mathematics 41
This can be
divided which
leaves the
radical in the
denominator.
We do not leave
radicals in the
denominator. So
we need to
rationalize.
10
5 1
2*
2
2
2
2
![Page 42: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/42.jpg)
Martin-Gay, Developmental Mathematics 42
12
3 3
2 3*
3
3
3 3
2 9
3 3
6
3
2
Reduce the fraction.
Rationalize
![Page 43: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/43.jpg)
Martin-Gay, Developmental Mathematics 43
Try it yourself. Simplify the following.
7 14
49
6
3y
x
98
7
7 2
7
77 2
7 7
7 14
7 14
How about with variables…
1)
2) x3
y y
3 yx
y y y
3
2
x y
y y
3x y
y y
3
2
x y
y
![Page 44: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/44.jpg)
Martin-Gay, Developmental Mathematics 44
12
3 4 * 3
3 4 2
4
5
4x
x
2
2
2x
x x
2 x
x x
2 x
x
3
13
3 18
4
x x
x
6
3 *3 2
2
x x x
x x
4
9 2
2x
2 5 3
6
3 28
12
xy x y
x
2 4 2
6
3 4 *7???
4 *3
xy x xy y
x
![Page 45: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/45.jpg)
Martin-Gay, Developmental Mathematics 45
Happy MONDAY !
Quick Check on Wedn. This week:
Will be on gradebook
*Standard Form: vertex point, x and y intercepts, sketch
*Vertex Form: vertex point, x and y intercepts, sketch
*Radicals: Simplify, add, subtr, multi, div (rationalization)
*Solving Radical equations
![Page 46: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/46.jpg)
Martin-Gay, Developmental Mathematics 46
Quick Check Tomorrow
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Martin-Gay, Developmental Mathematics 47
Self-Evaluation, Correction, Practice
1) Self Evaluation: Mark right or wrong with….
“CO” Conceptual Errors
i.e. wrong formulas, wrong solving steps
“CA” computational Errors
2) Write the actual correction work on separate paper
Turn in (1) and (2) today
3) Practice: use the other version
![Page 48: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/48.jpg)
Martin-Gay, Developmental Mathematics 48
Simplify
5 6 5 3
5 3 5 3
5 6
5 3
When radicals have binomial radicals in the denominator, multiply the numerator and denominator by the conjugate of the denominator to eliminate the radical in the denominator.
Simplify by distributing or using FOIL. then continue to completely simplify.
5 3 5 6 5 18
5 9
23
4
59
Be sure to simplify all numbers outside the radical if they all have common factors.
![Page 49: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/49.jpg)
Martin-Gay, Developmental Mathematics 49
Rationalizing Binomial Radicals
6
3 2
3 6 2 3
7
3 5
4 3
12 3 3 4 5 15
13
![Page 50: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/50.jpg)
Martin-Gay, Developmental Mathematics 50
Solve x(x + 6) = 30 by the quadratic formula.
x2 + 6x + 30 = 0
a = 1, b = 6, c = 30
)1(2
)30)(1(4)6(6 2
x
2
120366
2
846
This is Complex Number !!!!
The Quadratic Formula
Example
![Page 51: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/51.jpg)
Martin-Gay, Developmental Mathematics 51
The expression under the radical sign in the
formula (b2 – 4ac) is called the discriminant.
The discriminant will take on a value that is
positive, 0, or negative.
The value of the discriminant indicates two
distinct real solutions, one real solution, or no
real solutions, respectively.
The Discriminant
![Page 52: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/52.jpg)
Martin-Gay, Developmental Mathematics 52
Use the discriminant to determine the number and
type of solutions for the following equation.
5 – 4x + 12x2 = 0
a = 12, b = –4, and c = 5
b2 – 4ac = (–4)2 – 4(12)(5)
= 16 – 240
= –224
There are no real solutions.
The Discriminant
Example
![Page 53: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/53.jpg)
Martin-Gay, Developmental Mathematics 53
Solving Quadratic Equations
Steps in Solving Quadratic Equations
1) If the equation is in the form (ax+b)2 = c, use
the square root property to solve.
2) If not solved in step 1, write the equation in
standard form.
3) Try to solve by factoring.
4) If you haven’t solved it yet, use the quadratic
formula.
![Page 54: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/54.jpg)
Martin-Gay, Developmental Mathematics 54
Solve 12x = 4x2 + 4.
0 = 4x2 – 12x + 4
0 = 4(x2 – 3x + 1)
Let a = 1, b = -3, c = 1
)1(2
)1)(1(4)3(3 2
x
2
493
2
53
Solving Equations
Example
![Page 55: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/55.jpg)
Martin-Gay, Developmental Mathematics 55
Solve the following quadratic equation.
02
1
8
5 2 mm
0485 2 mm
0)2)(25( mm
02025 mm or
25
2 mm or
Solving Equations
Example
![Page 56: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/56.jpg)
Martin-Gay, Developmental Mathematics 56
2/7/17 Complex Number
![Page 57: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/57.jpg)
Martin-Gay, Developmental Mathematics 57
What imaginary
roots actually
look like on
a graph (which
is why we don’t
usually graph them)
![Page 58: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/58.jpg)
Martin-Gay, Developmental Mathematics 58
i
1 i
Before, is no solution.
But now a special symbol is assigned
so we can carry on the computation.
negative
![Page 59: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/59.jpg)
Martin-Gay, Developmental Mathematics 59
Examples 1i
16 16 1
4 i 4i
162 2 81 1
2 9 i
9 2i
![Page 60: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/60.jpg)
Martin-Gay, Developmental Mathematics 60
2i 1 1 i i 1 !ALWAYS
3i i i i 2i i 1i i
4i 2 2 1 1iiii i i 1
5
6
i
i
2 2
2 2 2 1 ...
iiiii i i i i
iiiiii i i i the pattern repeats
1 !i Always
![Page 61: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/61.jpg)
Martin-Gay, Developmental Mathematics 61
26 2 i i
Simplify
Divide the exponent by 4 (or 2) and look at the
remainder.
1
26 4=6 with remainder 2
Method
1
i26
13
2 13
2
26 2=13 ( 1) 1
Method
i
![Page 62: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/62.jpg)
Martin-Gay, Developmental Mathematics 62
a] 8 3i i 24i2
24 1
Simplify each expression.
24
b] 5 20 i 5 i 20Remember that 1 i
i2 100 110 10
![Page 63: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/63.jpg)
Martin-Gay, Developmental Mathematics 63
When adding or subtracting complex numbers,
combine like terms.
Ex: 8 3i 2 5i
8 2 3 5i i
10 2i
![Page 64: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/64.jpg)
Martin-Gay, Developmental Mathematics 64
Multiplication:
8 5i 2 3i
16 24i 10i 15i2
F O I L
1614i 15
3114i
![Page 65: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/65.jpg)
Martin-Gay, Developmental Mathematics 65
Complex Numbers Warm-Ups 8 minutes
30 45
2
:
}
b} (3 4 )
c} (3 4 )(3 4 )
d} (4 )(4 )
Multiply and Simplify
a i i
i
i i
i i
![Page 66: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/66.jpg)
Martin-Gay, Developmental Mathematics 66
5]
3Ex
12]
36ex
i
5
3
3
3
5 3
3
1
3i
i
i
23
i
i
3
i
3
i
18 3]
27
iTry
i
6
9
i
i
6
9
i i
i i
2
2
6 1 6
9 9
i i i
i
Division & Rationalizing the Denominator
Denominator can only
be an integer !
![Page 67: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/67.jpg)
Martin-Gay, Developmental Mathematics 67
Warm-up 8 minutes
202 55
:
]
] (4 5 )(4 5 )
] (2 6 )(2 6 )
Simplify
a i i
b i i
c
Solve using Quadratic Formula, what is the discriminant?
2] 2 2d y x x
![Page 68: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/68.jpg)
Martin-Gay, Developmental Mathematics 68
Radical expression and Exponents
Exponents v.s. Radicals (Roots)
35 125
3 125 5therefore
![Page 69: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/69.jpg)
Martin-Gay, Developmental Mathematics 69
Other examples
4
4 4
16
1
2 2 2 2
2 2
2
26 2 2
3
3 3
8
8
2 2 2
2 2 2
2
2
2
2 2
4
4
2
2 2
2 2
24
![Page 70: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/70.jpg)
Martin-Gay, Developmental Mathematics 70
2] 64a
3
4
5
6
7
b] 64
] 64
] 64
] 64
] 64
c
d
e
f
2 2 22 2 2 2 32 8
3
4
5
6
7
2 2 2
2 2 2 2
2 2 2 2 2
2 2
2 2
2
2 2 2 2 2
2 2 2 2
2 2
2
2
2
4
5
7
2 4
4
2
2
64
2
2
More Examples:
only 6 of them, not enough…
![Page 71: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/71.jpg)
Martin-Gay, Developmental Mathematics 71
2] 81a
3
4
5
b] 81
] 81
] 81
c
d
2 3 3 3 3 23 9
3
4
5
3
3 3 3
3 3 3
3 3 3
3
3
Not enough numbers
3
3
3 3
3
81still
More Examples:
![Page 72: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/72.jpg)
Martin-Gay, Developmental Mathematics 72
1
33
3
3
8 8 8 8 512
8 8 2 2 2 2
1
44 4
416 16 16 16 16 65536
16 16 2 2 2 2 2
2
6729Try
on calculator
9 ?did you get
Rational Exponents
16 ^ ( 14 )
Rational Exponent is
actually for calculators
yx
![Page 73: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/73.jpg)
Martin-Gay, Developmental Mathematics 73
Practice NOW!
103 240
2
] (2 4 )(2 7 )
] (2 3 )(2 3 )
] (2 3 )(2 3 )
]
] (2 8
1
)
:DOK Simplif
a i i
b i i
c
i
i
y
d i
e
2
2
2
2 :
] ( ) 2 8 3
] ( ) 6 10
? int?
y int? sk
]
e
] 2 3
2( 4) 6 8
tch
c
DOK
a f x x x
b g x x x
vert
d Solv
Solv
e
e x
x
x
e x
x
![Page 74: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/74.jpg)
Martin-Gay, Developmental Mathematics 74
Answers
] 24 22
]13
]1
] 1
] 60 32
a i
b
c
d i
e i
( 2, 5)
8 40int
4
0.42, 3.58
int (0,3)
vertex
x
y
(3,1)
6 4int
2
6 23
2
int (0,10)
vertex
x
ii
y
![Page 75: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/75.jpg)
Martin-Gay, Developmental Mathematics 75
2 53 2
ii
When you have a Binomial for Denominator:
3 23 2
ii
2
2
6 4 15 10
9 6 6 4
i i i
i i i
16 11
13i
6 7
6 7
24 4 7
36 6 7 6 7 7
24 4 7
29
4
6 7
16 1113 13
i
a.
b.
Rationalization for Binomials
![Page 76: Chapter 9: Quadratic Equations · 2017-02-12 · We previously have used factoring to solve quadratic equations. ... Another technique for solving quadratic equations is to use the](https://reader031.vdocuments.mx/reader031/viewer/2022021822/5b278d707f8b9aaa0d8b80ce/html5/thumbnails/76.jpg)
Martin-Gay, Developmental Mathematics 76
Ex: Solve x2+ 6x +10 = 0
2 4
2
b b acx
a
a =
b =
c =
26 6 4 1 10
2 1
1st
6 36 4 1 10
2 1
2nd 6 36 40
2
6 4
2
6 2
2
i
6 2 6 2
2 2
i iand
3 3i and i