Quadratic Equations
Solving a Quadratic Equation
• by factorization
• by graphical method
• by taking square roots
By factorization
01072 xx0)2)(5( xx
02__05 xorx2__5 xorx
roots (solutions)
Excercise
• By factorization find the roots of the below equation
4)32( 2 x
By graphical method
01072 xx
x
y
O
roots
By taking square roots
4)32( 2 x432 x
232 x52 x
5.2xA quadratic equation must contain two roots.
?
By taking square roots
4)32( 2 x
432 x
232 x
152 orx 5.05.2 orx
In general, a quadratic equation may have :
(1) two distinct (unequal) real roots
(2) one double (repeated) real root
(3) no real roots
Two distinct (unequal) real roots
x-intercepts
One double (repeated) real roots
x-intercept
No real roots
no x-intercept
Linear Functions and Their Graphs
c > 0
x
y
O
c < 0
x
y
O
c >0
x
y
O
m > 0
c
c <0 x
y
O
m > 0
c
c >0
x
y
O
m < 0
c
c <0
x
y
O
m < 0c
c =0
x
y
O
m < 0c
y = ax2
Draw the graph of the below function:
x
y
O
y = ax2
(a > 0)
Absolute Values
Let x be any real number. The absolute value of x, denoted by | x |, is defined as
xx if x 0.≧
-x if x < 0.
eg. | 5 | = 5, | 0 | = 0, | -5 | = 5
For all real numbers x and y,
xx 22
xx yxyx
y
x
y
x (y ≠ 0)
Generalization
If | x | = a, where a 0, ≧then x = a or x = -a
1) Fundamentals of Statistics by S.C.Gupta
2) Statistical and Quantitative Methods by Ranjeet Chitale
3) Statistics for Management by Levin and Rubin
4) Quantitative Techniques Vol1 and Vol2 by L.C.Jhamb
5) Quantitative Techniques – N.D.Vohra
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