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%$ Name : Subject : Chapter :
Mathematics 2 Quadratic Euati!ns "2#
Teacher: Class : Less! ss!n N! : $ate : Time :
Cheng Wui Leap F4 Maths 2 %&'%'%4 "Fri# %%(%)am*%(%)pm
Notes:1. Quadratic equations are euati!ns +hich ,ul,ill the ,!ll!+ing characteristics: "a# ha-e an equal ‘=’ sign "b# ha-e !nl. !ne un/n!+n "c# ha-e 2 as the highest p!+er !, the un/n!+n( 2( The general form !, a uadratic euati!n is +ritten as: "a# ax2 + bx 0 c = 0, +here a≠ 1 b ≠ 0 and c ≠ 1 e(g( 3x2 + 2x 0 & 3 1
"b# ax2 + bx = 0, +here a≠ 1 b ≠ 0 but 0 but c = 1 2 e(g( 2x + 5x 3 1
"c# ax2 + c = 0, +here a≠ 1 c ≠ 0 but 0 but b = 1 2 e(g( 3x + & + & 3 1
N!tes:* % Roots !, a uadratic euati!n are values !, the unknown +hich satisfy the uadratic euati!n( 2 T! determine +hether a gi-en -alue !, the un/n!+n is a r!!t !, a speci,ic uadratic euati!n substitute the gi-en -alue ,!r the un/n!+n int! the the euati!n( euati!n( , it satis,ies the euati!n then the -alue !, the un/n!+n un/n!+n is a r!!t !, the euati!n and -ice -ersa( 5 6!!ts !, an euati!n are als! called the solution !, an euati!n( There,!re in the ab!-e e7ample x e7ample x 3 * % and x and x 3 % are s!luti!ns !, the euati!n 3x2 0 4x + % 3 1(
E7ercise % 8sing the substituti!n meth!d determine +hether the gi-en gi-en -alues -alues !, the un/n!+ns un/n!+ns are r!!ts !, the respecti-e uadratic euati!ns( 2 19 x = −4 1 2 2 5 "a# x + x − %2 = 19
"b# p
2
"c# +
−
( h + %) ( h − 2)
=
1
: p − & = 19 p = − & 1% 2
"d# ( : x + %) ( ) x − 5) 31
2( $etermine the r!!ts !, the ,!ll!+ing uadratic euati!ns( "a# ( x − 2 ) ( x + %)
=
1
"b# ( 5 y − ) ) ( −: y − %)
21&51;&)(d!c
"e#
=
( −2d − 5) ( ;d − %)
",# m ( m − %)
1
2
=
1
=1
2 "b# p
"g# 5q ( q − 5)
=
=
5p + 4
1
"c# m ( m − ) )
"h# − x ( ) − x )
=
−
:=1
1
2 "d# y
=
)y − :
5( S!l-e the ,!ll!+ing uadratic euati!ns( "a# x = : x 2 − 2
"e# 2d 2 + 5d = 1
21&51;&)(d!c
5
",# x 2 − 4 = 1 "j# 2 x 2 + 5 x − 2 = 1
2 "g# q
=
2) 2 "/# 2 y − ) y + 2 = 1
"h# m 2
=
4m
"l# p
"i# %2 x 2 + %5x = 4
21&51;&)(d!c
2
−
4p + 4 = 1
2 "m# 5 g − %4 g + ; = 1
4
E7ercise %( S!l-e the ,!ll!+ing uadratic euati!ns( "a# 2 f
2
−&
"e#
2 x − & 5
=
x+: 5 x − 4
f − 4 = 1
",#
"b# )h 2 + %5h + : = 1
2 x − %
x − 5
=
)− x 2x − )
"c# :h 2 + & h = 21 2(
"d#
5
p + &
=
p 5p + )
21&51;&)(d!c
)
5(The length !, a pa-ement e7ceeds ,!ur times its breadth b. 5 m( The area !, the pa-ement is 52
% 2
5. Find the length !, the sides !, a rectangular card +h!se perimeter is 4 cm and area is %21 cm2(
suare metres( Find the
length and breadth !, the pa-ement(
The height !, a ball ab!-e the gr!undt sec!nds a,ter it is /ic/ed is gi-en b. the ,!rmula h 3 ;t *t 2 , +here h is the height in metres( Find the durati!n the ball is m!re than & metres ab!-e the gr!und=
4 The -alues !, t+! c!nsecuti-e p!siti-e numbers are such that the sum !, their suares is 2)( Find the numbers(
21&51;&)(d!c
! int: Time 3 $istance ÷ Speed#
7
%1(
The price !, a certain s+eet is increased b. 5 sen( F!r 6M%(;1 >assan recei-ed 2 s+eets less than the uantit. he recei-ed pre-i!usl.(What +as the !riginal price !, the s+eet=
< rectangular mirr!r %1 cm l!ng and ; cm +ide has a plastic b!rder !, c!nstant +idth ar!und it as sh!+n in the ab!-e diagram( , the area !, the mirr!r is 4; cm 2 +hat is the +idth !, its b!rder=
21&51;&)(d!c
&
11 The height !, a certain triangle is ) cm m!re than the length !, its base( , the area !, the triangle is 2) cm 2 ,ind "a# the length !, its base
"c#
"e#
−
4 5 %
5
% 2
4
−
) 2
)
5
"b# − −2 )
"d# *5) ",# 2
2( *% !r & 5(@readth 3
2
m Length 33 %5 m
4( %%%2 )( ; %) ( sec!nds &( @readth 3 %1 cm Length 3 %) cm ;( %) sen A( "a# %1 mh*% "b# 4mh *% %1( % cm %% "a# )cm "b# %1cm
"b# its height(
1" ?i-en that p = 4 is a r!!t !, the euati!n 6p2 - mp +p 3 1 +here m is a c!nstant ,ind the -alue !, m(
21&51;&)(d!c
)
;
% Fact!rise the ,!ll!+ing e7pressi!ns( "a# %; x2 B 2
"e# % * p * %2 p2
"b# x2 * 5 x * %1
",# x2 * %5 x 0
"c# 2m5n B ;mn "g# 5 x * 5 y 0 xy * y2
"d# 5 x5 * 4 x * 4 "h# % * t * t 2
21&51;&)(d!c
A
"i# %2 p2 * 4 p * %
"m# x2 * 5 x 0 xy * 5 y
"j# p"q * %# * "% * q#
"n# " y * # B " B y#
"/# 5 x5 * 2& "!# ) x2 * 4" x2 0 %#
"p# ) 0 x" x 0 # 2
"l# 2 p B l1 p 0 ;
21&51;&)(d!c
%1
2 Write the ,!ll!+ing uadratic 7 euati!ns in general ,!rm ax2 0 bx 0 c 3 1 (
2
"# m * 2m * %)
"a# m"m * )# 3
"b# " x * 5#" x 0 )# 3 2 x2
"r# "2 * x # 0 4"2 * x#
"s# 2)m * m5 "c# * k "k * 5# 3 4k
"t# k * k "k * %#
21&51;&)(d!c
%%
"e# )m2* "m * 4#2 3 %
21&51;&)(d!c
%2
%(
2(
21&51;&)(d!c
%5
