quadratic equations lesson 2 source : lial, hungerford and holcomb (2007), mathematics with...
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Quadratic equationsLesson 2
Source :Lial, Hungerford and Holcomb (2007), Mathematics with Applications, 9th edition, Pearson Prentice Hall, ISBN 0-321-44947-9 (Chapter 1 pp.56-64)
Quadratic Equations
Quadratic equations could be of two types:
1. Complete
Where a, b, and c are real numbers and a 0
2. Incomplete:
02 bxax 02 cax02 ax
02 cbxax
Quadratic Equations: Incomplete Quadratic Equations
Solving Incomplete Quadratic Equations:
02 bxax
3;0:
0)3(;02
0)3(2
062
:2
xxAnswer
xx
xx
xx
Example
Quadratic Equations: Incomplete Quadratic Equations
Solving Incomplete Quadratic Equations:
02 cax
2;2:
4
82
082
:
2
2
2
xxAnswer
x
x
x
Example
Quadratic Equations: Complete Quadratic Equations
Solving Complete Quadratic Equations:
02 cbxax
Three methods to solve:
1. Using Discriminant
2. Viete’s Theorem
3. Completing the square
Quadratic Equations: Complete Quadratic Equations
Solving Complete Quadratic Equations: 1.Using Discriminant:
02 cbxax
antdiscriacbD
a
acbbx
Solution
min4
2
4
:
2
2
2,1
3. Discriminant can be negative and we'd get no real solutions.
The "discriminat" tells us what type of solutions we'll have.1. Discriminant can be positive and we'd get two unequal real
solutions 04D 2 acb
2. Discriminant can be zero and we'd get one solution (called a repeated or double root because it would gives us two equal real solutions).
04D 2 acb
04 D 2 acb
Quadratic Equations: Complete Quadratic Equations
Example 3. Discriminant can be negative and we'd get no real solutions.
Feel the power of the formula!
Example 1. Discriminant can be positive and we'd get two unequal real solutions 08152 2 xx
Example 2. Discriminant can be zero and we'd get one solution (called a repeated or double root because it would gives us two equal real solutions).
0962 xx
053x2 x
Quadratic Equations: Complete Quadratic Equations
Quadratic Equations: Complete Quadratic Equations
028122 xx We will complete the square
First get the constant term on the other side28122 xx
___ 28___ 122 xx
We are now going to add a number to the left side so it will factor into a perfect square.
36 36 6436 122 xx
Solving Complete Quadratic Equations: 2. Completing Square
6436 122 xx 646 2 xThis can be written as
Now we'll get rid of the square by square rooting both sides.
646 2 x Remember you need both the positive and negative root!
86 x Add 6 to both sides to get x alone.
86 x
14861 x 2862 x
Quadratic Equations: Complete Quadratic Equations
011244 2 xx
Example: Solve by completing the square:
Quadratic Equations: Complete Quadratic Equations
2
1;
2
11:
5)62(5)62(
25)62(
25)62(
6116622)2(
11622)2(
11244
011244
21
2
222
2
2
2
xxAnswer
xandx
x
x
xx
xx
xx
xx
Solution:
Quadratic Equations: Complete Quadratic Equations
According to Viete’s theorem if x1 and x2 are the solutions of
then
Solving Complete Quadratic Equations: 3.Viete’s theorem
02 cbxax
Quadratic Equations: Complete Quadratic Equations
074392 xx
Example 1: Solve by Viete’s theorem:
Example 2: Solve by Viete’s theorem:
0210573 2 xx
Quadratic Equations: Complete Quadratic Equations
)37,2(:
39
74
07439
21
21
2
Answer
xx
xx
xx
Example 1: Solve by Viete’s theorem:
Quadratic Equations: Complete Quadratic Equations
Example 2: Solve by Viete’s theorem:
)5,14(:
193
57
703
210
0210573
21
21
2
Answer
xx
xx
xx
Solving by reducing to quadratic equations:
Example 1: Reduce to quadratic equation and solve
Practice yourself: Reduce to quadratic equation and solve
023 24 xx
Some higher degree equations could be reduced to quadratic equations and then solved
087 ) 36 xxa
084)5(8)5( ) 222 xxxxb
Solving by reducing to quadratic equations:Example 1: Solution
)2,2,1,1(x :Answer
2 2 2
1 1 1
,
3
2
023
:expression theRe
: variablenew Introduce
023
4321
3,22
22
2,12
11
21
21
2
2
24
xxx
xxyy
xxyy
Hence
yy
yy
yy
write
xy
xx
Important!If we know the roots of quadratic equation
we could factorize any quadratic expression easy way:
Example 2: Solve by Viete’s theorem:
Factorizing using roots
Example 1: Factorizing using roots
Solution:
028122 xx
141 x 22 x
Factorizing:
)2)(14())((2812 212 xxxxxxaxx
Factorizing using roots
Example 2: Factorizing using roots
Solution:
51 x 142 x
Factorizing:
)14)(153()14)(5(3
)14)(5(3))((210573 212
xxxx
xxxxxxaxx
0210573 2 xx