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Chapter 1 Quadratic Equations in One Unknown
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1.1 Real Number System
1.3 Solving Quadratic Equations by Quadratic
Contents1
1.2 Solving Quadratic Equations by Factor Method
Quadratic Equations in One Unknown
Formula
1.4 Solving Quadratic Equations by Graphical Method
1.5 The Nature of the Roots of Quadratic Equations
P. 2
Quadratic Equations in One Unknown1
Content
A. Integers
1.1 Real Number System
The numbers 1, 2, 3, 4,…usually used in counting are called natural numbers. The natural numbers and their negatives, such as –1, –2, –3, –4,…together with the number 0 are called integers.
B. Rational Numbers
Any number that can be expressed in the form (where p and q are
integers and q ≠ 0) is called a rational number, for example,
qp
)10
7(7.0,)
1
4(4),
1
0(0,)
1
3(3,)
5
21(
5
14,
19
11,
2
5,
3
2
1
nFurthermore, any integer n can be written as , which is a fraction. So, all integers are also rational numbers.
P. 3
Quadratic Equations in One Unknown1
Content
1.1 Real Number System
C. Characteristics of Rational Numbers in Decimals
A rational number that is not an integer can always be expressed as either a terminating decimal or a recurring (repeating) decimal.
Rational numbers whose decimals terminate
Rational numbers whose
decimals repeat
2.05
1 )1.0...(11111.0
9
1 i.e.,
25.04
1
375.08
3
)38.0...(83333.06
5 i.e.,
)742851.0...(571428571428.07
1 i.e.,
Table 1.1
P. 4
Quadratic Equations in One Unknown1
Content
1.1 Real Number System
D. Irrational Numbers
E. Real Numbers
Real numbers are either rational numbers or irrational numbers.The line is called a real number line and each point on the line represents a real number.
.32 πand , example, for , an called is
integers two of ratio the as expressed be cannot that numberAny
number irrational
Fig. 1.1
line. nubmer real the on points different twoby drepresente
are number irrational the and number rational the 1.1, Fig. In 2411.
P. 5
Quadratic Equations in One Unknown1
Content
1.2 Solving Quadratic Equations by Factor Method
A. Quadratic Equations
03413)1(4)1(.L.H.S 2
Now, if we put x = 1 into the equation (*), we have
and R.H.S. = 0
Therefore x = 1 satisfies equation (*).
Thus, 1 and 3 are called the roots of the equation (*).
Consider the equation
We call it a quadratic equation in one unknown because it is an equation of the second degree and it contains one unknown x.
).(........................................0342 xx
031293)3(4)3(L.H.S. 2
Similarly, putting x = 3 into the equation (*), we have
and R.H.S. = 0
Therefore x = 3 also satisfies equation (*).
P. 6
Quadratic Equations in One Unknown1
Content
B. Using the Factor Method to Solve a Quadratic Equation
For two real numbers u and v, if uv = 0, then u = 0 or v = 0.
The above method of solving quadratic equations is called factor method.
If a quadratic equation can be factorized into a product of two linear factors, then we can solve it by using the following fact:
3or 103or 01
0342
xxx
xx< Apply the above fact,
.0342 xxSolve
Example:
1.2 Solving Quadratic Equations by Factor Method
P. 7
Quadratic Equations in One Unknown1
Content
A. Completing the Square
A quadratic expression is called a perfect square if it can be factorized into a product of two identical linear factors.
The Method of Completing the Square
2
2
k
kxx 2By adding the term to , we have the following two
perfect squares.
22
2
22
kx
kkxx
22
2
22
kx
kkxxor
1.3 Solving Quadratic Equations by Quadratic Formula
P. 8
Quadratic Equations in One Unknown1
Content
B. Using Quadratic Formula to Solve a Quadratic Equation
.2
4
)0( 0
2
2
a
acbbx
acbxax
by given are, equationquadratic the of roots The
Quadratic Formula
1.3 Solving Quadratic Equations by Quadratic Formula
P. 9
Quadratic Equations in One Unknown1
Content
1.3 Solving Quadratic Equations by Quadratic Formula
a
cx
a
bx
a
cx
a
bx
acbxax
2
2
2
0
.0,0 where Solve
a
acb
a
bx
a
acb
a
bx
2
4
2
4
4
22
2
22
a
acbbx
2
42
Deriving the quadratic formula by completing the square
0 a
.expression side-left
the for square the Completing 22
2
22
a
b
a
c
a
bx
a
bx
P. 10
Quadratic Equations in One Unknown1
Content
:222 xxy function the of graph the Consider
Fig. 1.2
In Fig. 1.2, the curve cuts the x-axis at the points x = –2.7 and x = 0.7. These are the values of x when y = 0, that is x2 + 2x – 2 = 0.
The roots of a quadratic equation ax2 + bx + c = 0 (a 0) can be obtained by finding the x-intercepts of the graph of y = ax2 + bx + c.
When there is no x-intercept, the quadratic equation has no real root.
1.4 Solving Quadratic Equations by Graphical Method
P. 11
Quadratic Equations in One Unknown1
Content
.2
4
*).........(..........).........002
2
a
acbbx
acbxax
by given are (*) of roots the thatknow we1.3, Section In
( equationquadratic a Consider
.2
4
2
4
4),04(0
22
22
a
acbb
a
acbb
acbacb
and
has equation the So,
number. real positive a is then If 1: Case
:roots real unequal two
1.5 The Nature of the Roots of Quadratic Equations
P. 12
Quadratic Equations in One Unknown1
Content. has equation the So, number. real a not is then
,)(Δ If :3 Case
roots real noacb
acb
4
040
2
2
.2
4
*).........(..........).........002
2
a
acbbx
acbxax
by given are (*) of roots the thatknow we1.3, Section In
( equationquadratic a Consider
root. real double one has equation the Hence
. as simplified be can which, are equation the of roots the So,
zero. is then ,)(Δ If :2 Case
a
b
a
b
acbacb
22
0
4040 22
1.5 The Nature of the Roots of Quadratic Equations
P. 13
Quadratic Equations in One Unknown1
Content
.4Δ
2
2
cbxaxyxacb
of graph the of intercepts- of number the determine to use also can We
.)intercepts- two are there (i.e., points. distinct two at axis- the cuts of graph the then , If : 1 Case
xxcbxaxy 20Δ
0 16
)3)(1(4)2(
32
2
2
that Note
1.19(a). Fig. in shown as Consider :Example xxy
Fig. 1.19(a)
.4 )0( 22 acbacbxaxy and Let
1.5 The Nature of the Roots of Quadratic Equations
P. 14
Quadratic Equations in One Unknown1
Content
.intercept)-
oneonly is there (i.e.,only point one at axis-
the touches of graph the then ,Δ If : 2 Case
x
x
cbxaxy 20
).intercepts- no are there axis(i.e.,- the
cut not does of graph the then 0,Δ If :3 Case
xx
cbxaxy 2
Fig. 1.19(b)
Fig. 1.19(c)
that Note
1.19(b). Fig. in shown as Consider :Example
01616
)4)(1(4)4(
44
2
2
xxy
07
)2)(1(4)1(
2
2
2
that Note
).Fig.1.19(c in shown as Consider :Example
xxy
1.5 The Nature of the Roots of Quadratic Equations