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Quadratic Equation II

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Quadratic Equations,Quadratic Functions andAbsolute Values


Solving a Quadratic Equationby factorization by graphical methodby taking square rootsby quadratic equation by using completing square


By factorization

01072 xx0)2)(5( xx

02__05 xorx2__5 xorx

roots (solutions)


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


Solving a Quadratic Equation by the quadratic Formula


By quadratic equation

0,02 acbxaxIf

a

acbbx

2

42


)1(2

)10)(1(4)7()7( 2 x

01072 xx

a = b = c =1 10-7

25 xorx


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


Nature of Roots


△ = b2 - 4acSince the expression b2 - 4ac can be used to determine the nature of the roots of a quadratic equation in the form ax2 – bx + c = 0, it is called the discriminant of the quadratic equation.


Two distinct (unequal) real roots

x-intercepts

△ = b2 - 4ac > 0


One double (repeated) real roots

x-intercept

△ = b2 - 4ac = 0


No real roots

no x-intercept

△ = b2 - 4ac < 0


Solving a Quadratic Equation by Completing the Square


Solving a Quadratic Equation by Completing the Square

0342 xx

03)2

4()

2

4(4 222 xx

034)2( 2 x

1)2( 2 x

12 x

2121 orx31 orx


Relations between the Roots and the Coefficients


If α and β(p and q, x1 and x2) are the roots of ax2 + bx + c = 0,then sum of roots = α + β

a

b

and product of roots = αβ

a

c


Forming Quadratic Equations with Given Roots


x2 – (sum of the roots)x + (product of roots) = 0

Forming Quadratic Equations with Given Roots

In S.3, when α = 2 and β = -3

x = 2 or x = -3

x – 2 = 0 or x + 3 = 0

(x – 2)(x + 3) = 0

x2 + x – 6 = 0


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


Open upwards

Vertex

Open upwards

Line of symmetry

(a ＞0)


Open downwards

Vertex

Line of symmetry (a ＞0)


Vertex (Turning point)

Local (Relative) Maximum point (max. pt.)

Local (Relative) Minimum point point (mini. pt.)


y = ax2


x

y

O

y = ax2

(a ＞ 0)


x

y

O

y = ax2 + bx + c

(a ＞ 0)b2 - 4ac ＞ 0

roots

2 real roots(c ＜ 0)


x

y

O

(a ＞ 0)

b2 - 4ac ＝ 0

repeated roots

root

y = ax2 + bx + c

(c ＞ 0)


x

y

O

(a ＞ 0)

B2 - 4ac ＜ 0

No real roots

y = ax2 + bx + c

(c ＞ 0)

No intercept


Finding the turning point of a Quadratic Function by Completing the Square

0342 xx

03)2

4()

2

4(4 222 xx

034)2( 2 x

01)2( 2 x

mini value of the function = -1mini point = (-2, -1)

Because a = +ve, there exists a minimum point.


Absolute Values


Let x be any real number. The absolute value of x, denoted by | x |, is defined as x

x 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


Thank you

angstrom care 1 angstrom care quadratic equation ii

Documents

c slide

real roots slide

square slide

square roots slide

y o slide

given roots slide

nature of roots slide

quadratic equation slide