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Concept Category 4
Quadratic
Equations
§ 1
Solving Quadratic
Equations by the Square
Root Property
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
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
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
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
Martin-Gay, Developmental Mathematics 7
Solve (3x – 17)2 = 28
72173 x
3
7217 x
7228 3x – 17 =
Square Root Property
Example
§ 2
Solving Quadratic
Equations by the
Quadratic Formula
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.
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
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
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
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
Martin-Gay, Developmental Mathematics 14
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?
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
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
Martin-Gay, Developmental Mathematics 18
4
16
25
100
144
= 2
= 4
= 5
= 10
= 12
Radicals
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
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
Martin-Gay, Developmental Mathematics 21
Simplify each expression
737576 78
62747365 7763
A]
B]
Martin-Gay, Developmental Mathematics 22
+ To combine radicals: combine the
coefficients of like radicals
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
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
Martin-Gay, Developmental Mathematics 25
* Multiply the coefficients and then
multiply the radicands and then simplify
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
Martin-Gay, Developmental Mathematics 27
2X
6Y
264 YXP
244 YX
10825 DC
= X
= Y3
= P2X3Y
= 2X2Y
= 5C4D5
Martin-Gay, Developmental Mathematics 28
3X
XX
=
=
XX *2
YY 45Y
=
= YY 2
Martin-Gay, Developmental Mathematics 29
33YPX
2712 YX
9825 DC
=
=
= 5Y
PXYYX *22
5Y
PXYXY=
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
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
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
Martin-Gay, Developmental Mathematics 33
2 2
3 5 4 20 3 45 4 3
Practice NOW
4
43 2 2 16x x
Martin-Gay, Developmental Mathematics 34
SOLUTIONS
2
3 5
316x
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!
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
Martin-Gay, Developmental Mathematics 37
7
568 2*4 22
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
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.
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
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
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
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
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
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
Martin-Gay, Developmental Mathematics 46
Quick Check Tomorrow
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
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.
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
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
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
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
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.
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
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
Martin-Gay, Developmental Mathematics 56
2/7/17 Complex Number
Martin-Gay, Developmental Mathematics 57
What imaginary
roots actually
look like on
a graph (which
is why we don’t
usually graph them)
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
Martin-Gay, Developmental Mathematics 59
Examples 1i
16 16 1
4 i 4i
162 2 81 1
2 9 i
9 2i
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
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
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
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
Martin-Gay, Developmental Mathematics 64
Multiplication:
8 5i 2 3i
16 24i 10i 15i2
F O I L
1614i 15
3114i
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
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 !
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
Martin-Gay, Developmental Mathematics 68
Radical expression and Exponents
Exponents v.s. Radicals (Roots)
35 125
3 125 5therefore
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
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…
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:
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
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
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
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
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
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