module 2 - quadratic equation & quadratic functions

MODULE 2

QUADRATIC EQUATIONS & QUADRATIC

MODULPROGRAM IBNU SINAADDITIONAL MATHEMATICS

Terbitan :-YAYASAN PELAJARAN JOHORJABATAN PELAJARAN NEGERI JOHOR

QUADRATIC EQUATIONS AND QUADRATIC FUNCTIONS FORM 4

MODULE 2IBNU SINA

TOPIC : QUADRATIC EQUATIONS & QUADRATIC FUNCTIONS

Express Note :

General form

bx + c = 0 , where a , b and c are constants , a 0

Properties

1. Equation must be in one unknown only

2. The highest power of the unknown is 2

Examples

1.2x 2 + 3x – 1 = 0 is a quadratic equation

2.4x 2 – 9 = 0 is a quadratic equation

3.8x 3 – 4x2 = 0 is not a quadratic equation

Determining roots of a quadratic equation

i. factorisation

ii completing the square

iii. formula x =

Forming QE from given roots by expansion

SOR = , POR =

x 2 - (SOR)x + (POR) = 0

Types of roots for QE

i. two distict / different roots : b2-4ac > 0

ii. two equal roots : b2-4ac = 0

iii. no roots : b2-4ac < 0

PROGRAM IBNU SINA TAHUN 2010(ALL A’s)


Shapes of graph of Quadratic Function

[ f(x) = ax 2 + bx + c ]

If a > 0 If a<0 mak *

* min

Relation between the position of Quadratic Function Graphs and its roots

i. graph intersects the x-axis at two points

b2-4ac > 0

ii graph does not intersects the x-axis

b2-4ac < 0

iii graph touches the x-axis at one point

b2-4ac = 0

iv graph touches the x-axis

b2-4ac ≥ 0

Finding the min @ max value of QF using the completing the square method

f(x) = ax 2 + bx + c General form

= a(x + p )2 + q After completing the square

If a>0 min vakue = q

axis of symmetry : x = -p

min point = (-p, q)

If a<0 nilai max value = q

axis 0f symmetry : x = -p

mak point. = (-p, q) ]

Sketching Quadratic Function Graph

i Determine the shape (identify a)

ii Determine the position (evaluate b2-4ac)



iii. Completing the square (find max@ min point and axis of symmtery)

iv. Solve f(x) = 0 (find point of intersection with x-axis)

v. Find f(0) (find point of intersection with y-axis)

vi Plot the points and connect them with a smooth curve

Quadratic Inequalities

Range of Quadratic Inequalities

Using line number

i. Factorise

ii State two values of x

ii Use suitable method to find the correct position / area

iii State the range

a b

x < a x > b

a < x < b

PAPER 1

1. Solve the following quadratic equations. Give your answer correct to 4 significant

figures.



(a) x2 + 7x + 1 = 0

(b) x( x + 5) = 5.

(c) x(2 + x) = 10

(d) 2x(2 – x) = 2x – 1

2. Express 2x2 – 5x – 2 = 0 in the form of a(x – p)2 + q = 0. Hence, solve the quadratic

equation .

3.(a) Given that α and β are the roots of the quadratic x2 + 4x + 7 = 0. Form a quadratic

equation with roots α – 1 and β – 1 .

(b) Given that α and β are the roots of the quadratic 2x2 + 5x + 10 = 0. Form a quadratic

equation with roots α + 1 and β + 1 .

(c ) Given that α and β are the roots of the quadratic 3x2 - 6x + 1 = 0. Form a quadratic

equation with roots and .

4.(a) Given the quadratic equation 2x2 – 6x = 3px2 + p . Find the range of values of p if

the quadratic equation has two distinct roots.

(b) Given the quadratic equation 4x2 + p = 3(2x – 1). Find the range of values of p if the

quadratic equation has no roots.

(c) Given the quadratic equation x2 = 2(2-m)x + 4 - m2. Find the range of values of p if

the quadratic equation has two different roots.

5.Find the values of k if the quadratic equation has two equal

roots.

6.(a)Given that 5 and are the roots of the quadratic equation .

Find the value of m and n.



(b)Given that 3 and are the roots of the quadratic equation .

Find the value of m and n.

(c) Given that -2 and 6 are the roots of the quadratic equation . Find

the value of k and n.

7. One of the roots of quadratic equation is reciprocal of the other

root. Find the values of k and the roots of the quadratic equation.

8. Form the quadratic function in the form of ,

hence, find the minimum or maximum point.

9. Find the minimum or maximum point of the graph of quadratic function

.

10.(a)

10(b)

The diagram shows a graph of quadratic function , where p and

q are constant. Find the value of p, q and k. Hence, state the equation of the axis of

symmetry of the graph.



The diagram shows a graph of quadratic function , where p and

q are constant. Find the value of p, q and n.

11. Find the range of x in each of the following

(a)

(b)

(c) and f(x) is always positive.

(d)

12. Find the range of x if the quadratic function never

touches the x-axis.



PAPER 2

1.

y

x

0 A(3,m)

B

Diagram 1 shows the curve of a quadratic function f(x) = -x2 – nx +4. The curve has a

maximum point at A(3,m) and intersects that f(x)-axis at point B.

(a) State the coordinates of point B.

(b) By using the method of completing the square , find the value of m and n.

2. y

x 0

Q(k, m)

Diagram 2 shows the curve of a quadratic function f(x) = x2 + 5x - 3. The curve has a

minimum point at Q(k, m). By using the method of completing the square , find the

value of k and m.

END OF MODULE 2



MODULE 2 - ANSWERSTOPIC : QUADRATIC EQUATIONS AND QUADRATIC FUNCTIONS

PAPER 1

1. a)

or

b)

or

c)

or

d)

or

2.

or

3.(a)



(b)

(c )

4.(a)

(b)

(c )

(-6) 2-4(4)(p-3)<0

p>

(2m-4)2 -4(1)(m2-4)>0m>2

5.

6.(a)

(b)

S.O.R =

m = 10

P.O.R =

n = 13

S.O.R =

m =

P.O.R =

n =



(c)

S.O.R = -2 + 6 =

k = -20

P.O.R =

n =

7.roots =

S.O.R

roots = 3.186 and 0.3139

P.O.R

8.

minimum point ,

9.f(x) =

minimum point ;

10.(a)

(b)

p = 2 ; q = 5k = - 3

p = -2, q = 3,n = -9

11. a) and

b) and

c) and



d)

12.

PAPER 2

1.

2.

(a)(b)

B(0,4)-(x2 + nx – 4)m = 13, n= -6

k= , m =

END OF MODULE 2


module 2 - quadratic equation & quadratic functions

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