MODUL PROGRAM IBNU SINA ADDITIONAL MATHEMATICS
MODULE 2 QUADRATIC EQUATIONS & QUADRATIC FUNCTIONS
Terbitan :YAYASAN PELAJARAN JOHOR JABATAN PELAJARAN NEGERI JOHOR
QUADRATIC EQUATIONS AND QUADRATIC FUNCTIONS
FORM 4
MODULE 2 IBNU SINA TOPIC : QUADRATIC EQUATIONS & QUADRATIC FUNCTIONS
Express Note : General form ax
2
+
bx + c = 0
, where a , b and c are constants , a
≠
0
Properties 1.
Equation must be in one unknown only The highest power of the unknown is 2
2.
Examples
1.
2 x 2 + 3 x – 1 = 0 is a quadratic equation
2.
4 x 2 – 9 = 0 is a quadratic equation
3.
8 x 3 – 4 x 2 = 0 is not a quadratic equation
Determining roots of a quadratic equation i.
factorisation
ii
completing the square
iii.
formula x =
−b ±
b
2
− 4ac
2a
Forming QE from given roots by expansion
SOR =
−b a
, POR =
c a
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)
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QUADRATIC EQUATIONS AND QUADRATIC FUNCTIONS
FORM 4
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
If a>0
After completing the square
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) ]
PROGRAM IBNU SINA TAHUN 2010(ALL A’s)
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QUADRATIC EQUATIONS AND QUADRATIC FUNCTIONS
FORM 4
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
x>b a
PROGRAM IBNU SINA TAHUN 2010(ALL A’s)
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QUADRATIC EQUATIONS AND QUADRATIC FUNCTIONS
FORM 4
PAPER 1 1.
Solve the following quadratic equations. Give your answer correct to 4 significant figures.
(a) x 2 + 7 x + 1 = 0 (b) x ( x + 5) = 5. (c) x (2 + x ) = 10 (d) 2 x (2 – x ) = 2 x – 1 Express 2 x 2 – 5x – 2 = 0 in the form of a( x – p)2 + q = 0. Hence, solve the quadratic
2.
equation x
3.(a)
2
=
5 x + 2 . 2
Given that α and β are the roots of the quadratic x 2 + 4 x + 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 2 x 2 + 5 x + 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 3 x 2 - 6 x + 1 = 0. Form a quadratic equation with roots
4.(a)
α
2
and
β 2
.
Given the quadratic equation 2 x 2 – 6 x = 3px2 + p . Find the range of values of p if the quadratic equation has two distinct roots.
(b)
Given the quadratic equation 4 x 2 + p = 3(2x – 1). Find the range of values of p if the quadratic equation has no roots.
(c)
Given the quadratic equation x 2 = 2(2-m)x + 4 - m 2. 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
2
kx
+ kx − 5 x +
k
9
=0
has two equal
roots. 6.(a)
Given that 5 and
−
5 3
are the roots of the quadratic equation 3 x 2
PROGRAM IBNU SINA TAHUN 2010(ALL A’s)
+ mx − 2n + 1 = 0 .
Page 5
QUADRATIC EQUATIONS AND QUADRATIC FUNCTIONS
FORM 4
Find the value of m and n. (b)
Given that 3 and
3 4
are the roots of the quadratic equation 2 x 2
+ 3mx − 7n + 2 = 0 .
Find the value of m and n. (c)
Given that -2 and 6 are the roots of the quadratic equation 5 x 2
+ kx + 8n −1 = 0 .
Find the value of k and n. 7.
One of the roots of quadratic equation kx 2 − (k + 5)x + 2 = 0 is reciprocal of the other root. Find the values of k and the roots of the quadratic equation.
8.
Form the quadratic function f ( x ) = 3 x 2 − 9 x + 4 in the form of f ( x ) = a( x − p ) 2
+q ,
hence, find the minimum or maximum point. 9.
Find the minimum or maximum point of the graph of quadratic function f ( x )
= 2 x 2 + 3 x − 6 .
10. (a)
The diagram shows a graph of quadratic function f ( x )
2( x − p ) 2
=−
+
q , 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.
10(b)
PROGRAM IBNU SINA TAHUN 2010(ALL A’s)
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QUADRATIC EQUATIONS AND QUADRATIC FUNCTIONS
The diagram shows a graph of quadratic function f ( x)
FORM 4
= −3( x + p) 2 + q , 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) ( x + 3)( 2x − 5) ≥ 0 (b)
( 2 x + 3 )( x − 5 ) ( x − 4 )
<0
(c) f ( x ) = 3 x 2 − 2x −16 and f ( x ) is always positive. (d) ( x −3)( 3 x +1) ≤ 0 12.
Find the range of x if the quadratic function f ( x)
= kx 2 +(2k +1) x +k −3 never
touches the x -axis.
PAPER 2
PROGRAM IBNU SINA TAHUN 2010(ALL A’s)
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QUADRATIC EQUATIONS AND QUADRATIC FUNCTIONS
FORM 4
1. y
x 0
A(3,m)
B
Diagram 1 shows the curve of a quadratic function f(x) = -x 2 – nx +4. The curve has a maximum point at A(3,m) and intersects that f(x)-axis at point B.
2.
(a)
State the coordinates of point B.
(b)
By using the method of completing the square , find the value of m and n. y
x 0
Q(k, m)
Diagram 2 shows the curve of a quadratic function f(x) = x 2 + 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
PROGRAM IBNU SINA TAHUN 2010(ALL A’s)
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QUADRATIC EQUATIONS AND QUADRATIC FUNCTIONS
FORM 4
MODULE 2 - ANSWERS TOPIC : QUADRATIC EQUATIONS AND QUADRATIC FUNCTIONS PAPER 1 1.
a)
x = x
b)
2
d)
or x
= −6.854
5
− 4(1)( −5)
2
2
or x = −0.5854
+ 2 x − 10 = 0
x =
x
−5 ±
= 0.8541
2
x
2
+ 5 x − 5 = 0
x =
c)
− 4(1)(1)
72
= −0.1459
x
x
−7 ±
−2 ±
2
− 4(1)( −10 )
2
2
=
2.31 or x
= −4.317
− 2 x 2 + 2 x + 1 = 0 − 2 ± 2 2 − 4( −2)(1) x = 2(2)
x
2.
or x
= 1.366
2
5 57 = 0 2 x − − 4 8 x
3.(a)
= −0.3660
= 3.137
or x = −0.6375
α + β = −4, αβ = 7 (α −1) + ( β −1)
= −6
(α −1)( β −1) x 2
+ 6 x +12
=12 =0
5
α + β = − , αβ = 5 2
(α +1) + (β +1)
=−
(α −1)( β −1)
7
2 x
2
=
1 2
2
+ x + 7 = 0
(b)
PROGRAM IBNU SINA TAHUN 2010(ALL A’s)
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QUADRATIC EQUATIONS AND QUADRATIC FUNCTIONS
α + β = −2, α β = α 2 (
β
+
α β 2
6 x
2
2
1 3
= −1
2
)(
FORM 4
)
=
1 6
+ 6 x + 1 = 0
(c ) 4.(a)
(−6) p
(b)
(c ) 5.
6.(a)
<
2
− 4(2)( p − 3) > 0
15 2
(-6) 2-4(4)(p-3)<0 21 p> 4 (2m-4)2 -4(1)(m2-4)>0 m>2 k − 4(k ) = 0 9 k 2 − 18 k + 45 = 0 k = 15; k = 3
(k − 5)
2
5 m = 3 3
S.O.R = 5 + − m = 10
5 2n − 1 = − 3 3
P.O.R = 5 × − n = 13
3 = − 3m 4 2
S.O.R = 3 + (b) m=
5
−
2 3
P.O.R = 3 ×
= 4
n=
− 7n + 2 2
5
−
14
PROGRAM IBNU SINA TAHUN 2010(ALL A’s)
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QUADRATIC EQUATIONS AND QUADRATIC FUNCTIONS
S.O.R = -2 + 6 =
−
FORM 4
k
5
k = -20
(c)
− 2 ×6 =
P.O.R =
n=
7.
roots = a;
8n −1 5
−59 8
1 a
S.O.R 1 5 = a+ a 2 2 2a − 5a + 2 = 0 1 a = ;2 2 roots = 3.186 and 0.3139
8.
P.O.R 1 a
×
a
=2
k
∴ k = 2
4 = 3 x 2 − 3 x + 3 2 3 11 = 3 x − − 2 4 3 11 minimum point , ,− 2 4 f ( x )
9.
+ 3 x − 3 2 2 3 57 2 x + − 4 8 2 x 2
f ( x ) =
minimum point ;
− 3 ,− 57 4 8
10.(a) p = 2 ; q = 5 k = - 3 (b)
11.
p = -2, q = 3, n = -9 a) b) c)
x ≤
−3
x <
−
x
and x ≥
3 and 2
< −2
4
5 2
< x <
and x >
5
8 3
PROGRAM IBNU SINA TAHUN 2010(ALL A’s)
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QUADRATIC EQUATIONS AND QUADRATIC FUNCTIONS
( x −3)( 3 x +1)
d)
−1 3
12.
FORM 4
≤0
≤ x ≤ 3
( 2k +1) 2 − 4k ( k −3)
<0
16 k +1 < 0 x
>
−1 16
PAPER 2 1.
2.
(a) (b)
B(0,4) -(x2 + nx – 4) m = 13, n= -6 k=
5
−
2
, m=
−37 4
END OF MODULE 2
PROGRAM IBNU SINA TAHUN 2010(ALL A’s)
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