Form 4
Chapter 2 – Quadratic Expressions and Equations
1. Qua Quadr drati aticc Ex Expr pres essio sions ns
(A) Identifying quadratic expression 1. A quadratic expression is expression is an algebraic expression of the form
ax2 + bx bx + +cc,
where a, b and and c c are are constants, a ≠ 0 and x and x is is an unknown. of x is is 2. (a) The highest power of x (b) or example, ! x2 " # x + $ is a quadratic expression.
Example 1
%tate whether each of the following is a quadratic expression in one unknown. &a' x &a' x2 " ! x x + + $ &b' ( p2 + )0 &c' ! x x + + # &d' 2 x2 + * y y + + )* &e' 2p+)p+#2p+)p+# &f' y &f' y$ " $ y + )
Solution:
(a) Yes. A quadratic expression in one unknown.
(b) Yes. A quadratic quadratic expression in one unknown.
1
Form 4
Chapter 2 – Quadratic Expressions and Equations
(c) Not a quadratic expression in one unknown. The highest power of the unknown x is not 2.
(d) Not a quadratic expression in one unknown. There are 2 unknowns, x and in the quadratic
expression.
(e) Not a quadratic expression in one unknown. The highest power of the unknown x is not 2.
)pp-))pp-)
(f) Not a quadratic expression in one unknown. The highest power of the unknown x is not 2.
. A quadratic quadratic expression can be formed b multipling two linear expressions. expressions.
&2 x x + + $'& x x $' 2 x2 " $ x x " " /
Example
ultipl the following pairs of linear expressions. &a' &* x x + + $'& x " x " 2' &b' & y y " " #'2 &c' 2 x x & & x " x " !'
2
Form 4
Chapter 2 – Quadratic Expressions and Equations Solution:
(a) &* x x + + $'& x " x " 2'
&* x x'& '& x' x' + &* x x'&2' '&2' +&$'& x x'' + &$'&2' * x2 " ( x x + + $ x x " " # * x2 " ! x x " " #
y " #'2 (b) & y " & y " y " #'& y " y " #' & y'& y'& y' y' + & y'&#' y'&#' + &#'& y y'' + &#'&#' y2 # y y " " # y y + + $# y2 )2 y y + + $#
x & & x " x " !' (c) 2 x 2 x x&& x' x' + 2 x x&!' &!' 2 x2 " )0 x
Example 11
orm a quadratic expression b multipling each of the following. &a'
&# p p " " 2'&2 p p " " )'
&b' &m &m + !'&* " m m' &c'
& x + x + 2' &2 x x " " $'
3
Form 4
Chapter 2 – Quadratic Expressions and Equations Solution:
&a' &# p p " " 2'&2 p p " " )' &# p p'&2 '&2 p p'' + &# p p'&)' '&)' + &2'&2 p p'' +&2'&)' )2 p2 " # p p " " * p p + + 2 )2 p2 " )0 p p + + 2
&b' &m &m + !'&* " m m' &m &m'&*' + &m &m'& '&m m' + &!'&*' + &!'&m &!'& m' *m *m " m m2 + 20 " $!m $!m " m m2 " $)m $)m + 20
&c' & x + x + 2' &2 x x " " $' & x'&2 x'&2 x x'' + & x'&$' x'&$' + &2'&2 x x'' + &2'&$' 2 x2 $ x x + + * x x " " # 2 x2 + + x x #
4
Form 4
Chapter 2 – Quadratic Expressions and Equations
. !ac !actori torisati sation on "f "f Quadr Quadratic atic Expr Express ession ion (A) !actorisation quadratic expressions of t#e form ax $ bx $ c% b & ' or c & '
1. !actorisation of quadratic expressions is a process of finding two linear expressions whose
product is the same as the quadratic expression. bx that that consist of two terms can be factorised b . 3uadratic expressions ax2 + c and ax2 + bx finding the common factors for both terms.
Example 1
actorise each of the following1 (a) 2 x2 + # (b) p2 " $ p (c) # x2 " / x
Solution: x $ ) 4 &2 is common factor' (a) 2 x2 + # ( x
(b) p2 " $ p p p (* p + ) 4 & p is p is common factor' (c) # x2 " / x ( x + ) 4 &$ x x x ( x is is common factor'
5
Form 4
Chapter 2 – Quadratic Expressions and Equations
(,) !actorisation of quadratic expressions in t#e form ax + c % -#ere a and c are perfect squares
Example (a) / p2 " )# (b) 2! x2 " ) (c) )*-)2!x2)*-)2!x2
Solution:
(a) / p2 " )#
&$ p p''2 " *2 &$ p p " " *' &$ p + *'
(b) 2! x2 " )
&! x x''2 " )2 &! x x " " )' &! x + )'
(c) )*-)2!x2
&)2'2 - &)!x'2 &)2-)!x'&)2+)!x')*-)2!x2 &)2'2-&)!x'2 &)2-)!x'&)2+)!x'
6
Form 4
Chapter 2 – Quadratic Expressions and Equations
() !actorisation quadratic expressions in t#e form ax $ bx $ c% -#ere a / '% b / ' and c / '
Example
actorise each of the following y " " ( (a) $ y2 + 2 y (b) * x2 " )2 x + /
Solution:
(a) !ac !actori torise se using using t#e t#e ross 0et#od
$ y2 + 2 y y " " ( ( y + ) ( y $ )
7
Form 4
Chapter 2 – Quadratic Expressions and Equations
(b)
+ ) ( x + + ) * x2 " )2 x + / ( x +
$. Quadratic Equations 1. Quadratic equations are equations which fulfill the following characteristics1 (a) 5a6e an equal 78 sign (b) 9ontain onl one un2no-n (c) 3ig#est po-er of the unknown is 2.
For example%
8
Form 4
Chapter 2 – Quadratic Expressions and Equations
. The general form of a quadratic equation is written as1 (a) ax $ bx $ $ c & ',
where a ≠ 0, b ≠ 0 and c ≠ 0, example1 * x2 + )$ x x " " )2 0
(b) ax $ bx & & ',
where a ≠ 0, b ≠ 0 but c & ', example1 x2 + / x x 0
(c) ax $ c & ',
where a ≠ 0, c ≠ 0 but b & ', example1 / x2 " $ 0
9
Form 4
Chapter 2 – Quadratic Expressions and Equations
Example 11
:ritee each quadratic equation in the general form. :rit &a' x &a' x2 " ! x )2 &b' 2 + ! x2 " # x x 0 &c' p2 " $ p p * p2 + * p p " " $ &d' & x x " " 2'& x + x + #' 0 &e' $ " )$ x x * & x x2 + 2' &f' 2-)-$2-)-$ &g' p*2p2-$)0p*2p2-$)0 &h' 2+!*-)22+!*-)2 &i' *pp&p-#'*pp&p-#'
Solution:
$ c & ' A quadratic equation in the general form is written as ax $ bx $
(a) x2 " ! x )2
x2 " ! x )2 0
x 0 (b) "2 + ! x2 " # x ! x2 " # x x "2 "2 0
10
Form 4
Chapter 2 – Quadratic Expressions and Equations
(c) p2 " $ p p * p2 + * p p " " $
p2 " $ p p " " * p2 " * p p + + $ 0 $ p2 " p + $ 0
(d) & x " x " 2'& x + x + #' 0
x2 + # x x " " 2 x x " " )2 0 x2 + * x x " " )2 0
(e) $ " )$ x x * & x x2 + 2'
$ " )$ x x * x2 + ( "* x2 " ( + $ " )$ x x 0 "* x2 " )$ x x " " ! 0 * x2 + )$ x x + + ! 0
(f) 2-)-$2-)-$
2 y y " " y y2 ) " $ y 2 y y " " y y2 " ) + $ y 0 " y " y2 + $ y y " " ) 0 y2 " $ y + ) 0
11
Form 4
Chapter 2 – Quadratic Expressions and Equations
(g) p*2p2-$)0p*2p2-$)0
)0 p p ( p2 " )2 "( p2 + )0 p +)2 0 ( p2 " )0 p p " " )2 0
(#) 2+!*-)22+!*-)2
22 + )0 * " * 22 " * + )0 + * 0 22 " * + )* 0
(i) *pp&p-#'*pp&p-#'
* p p p p & & p p " " #' * p p */ p2 " *2 p " */ p2 + *2 p + * p 0 */ p2 " *# p p 0
*. 4oots of Quadratic Equations
1. A root of quadratic equation is the 6alue of the unknown u nknown which satisfies the quadratic
equation. . ;oots of an equation are also called the solution of an equation. . To sol6e a quadratic equation b the factorisation method, follow the steps below1
12
Form 4
Chapter 2 – Quadratic Expressions and Equations Step 1:
expressions, that is, &mx &mx + + p p'' &nx &nx + + q' 0. Step 3:
Example 11
%ol6e the quadratic equation
13
Form 4
Chapter 2 – Quadratic Expressions and Equations
Solution:
14
Form 4
Chapter 2 – Quadratic Expressions and Equations
Example 1
%ol6e the quadratic equation * x2 " )2 ")$ x
Solution:
* x2 " )2 ")$ x * x2 + )$ x x " " )2 0 &* x x " " $'& x + x + *' 0 * x x " " $ 0, x$>* or x or x + + * 0 x x "*
Example 1
%ol6e the quadratic equation ! x2 $ & x + x + 2' " *
15
Form 4
Chapter 2 – Quadratic Expressions and Equations Solution:
! x2 $ & x + x + 2' " * ! x2 $ x x + + # " * ! x2 " $ x x " " 2 0 &! x + 2'& x " x " )' 0 ! x + 2 0, x-2>! or x or x " " ) 0 x x )
Example 1
%ol6e the quadratic equation
16
Form 4
Chapter 2 – Quadratic Expressions and Equations Solution:
Quadratic Equations 5ong Questions (Question 1 6 )
Question 11
%ol6e the quadratic equation, & y + y + $'& y " y " *' $0
Solution:
& y + y + $'& y y " " *' $0 y2 " * y y + + $ y y " " )2 $0 y2 " " y y " " )2 " $0 0 y2 " " y y " " *2 0 & y + y + #'& y y " " ' 0
17
Form 4
Chapter 2 – Quadratic Expressions and Equations
y + y + # 0, y 0, y "# ?r y " y " 0 y
Question 1
%ol6e the quadratic equation, ! x2 $& x $& x " " 2' + (
18
Form 4
Chapter 2 – Quadratic Expressions and Equations Solution:
! x2 $ x x " " # + ( ! x2 " $ x x " " 2 0 &! x x + + 2'& x " x " )' 0 ! x + 2 0, x 0, x -2>! ?r x " x " ) 0 x x )
Question 1
%ol6e the quadratic equation
19
Form 4
Chapter 2 – Quadratic Expressions and Equations Solution:
Question 1
%ol6e the quadratic equation
20
Form 4
Chapter 2 – Quadratic Expressions and Equations Solution:
Quadratic Equations 5ong Questions (Question 7 6 8)
Question 71
%ol6e the equation1 &m + 2'&m 2'&m " *' &m &m " *'.
Solution:
&m + 2'&m 2'&m " *' &m &m " *'
21
Form 4
Chapter 2 – Quadratic Expressions and Equations
m2 " *m *m + 2m 2m " ( m m " 2( m2 " /m /m + 20 0 &m " !'&m !'&m " *' 0 m & 7
or
m &
Question 91
%ol6e the equation1
Solution:
Question *1
%ol6e the equation1
22
Form 4
Chapter 2 – Quadratic Expressions and Equations
23
Form 4
Chapter 2 – Quadratic Expressions and Equations Solution:
Question 81
@iagram abo6e shows a rectangle ABCD rectangle ABCD.. (a)
24
Form 4
Chapter 2 – Quadratic Expressions and Equations Solution:
(a)
Area of ABCD of ABCD &n &n + ' B n &n &n2 + n n' cm2
(b)
i6en the area of ABCD of ABCD #0 n2 + n n #0 n2 + n n " #0 0 &n " !' &n &n + )2' 0 n!
or
n " )2 ¬ accepted'
:hen n !, Cength of AB of AB ! + 1 cm
25
