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