Quadratic Equations

2.1 Recognising Quadratic Equations

2.2 The ROOTs of a Quadratic Equation (Q.E) 2.3 To Solve Quadratic Equations

ax 2  bx  c  0

2.4 To Form Quadratic Equations From Given Roots

2.5 Relationship between

b 2  4ac and the roots of Q.E

2.1 Recognising Quadratic Equations

Students will be taught to 1. Understand the concept quadratic equations and its roots.

Students will be able to: 1.1 Recognise quadratic equation and express it in general form

QUADRATIC EQUATIONS (i) The general form of a quadratic equation is are constants and a ≠ 0.

ax 2  bx  c  0

(ii) Characteristics of a quadratic equation: (a) Involves only ONE variable, (b) Has an equal sign “ = ” and can be expressed in the form ,

ax 2  bx  c  0 (c) The highest power of the variable is 2.

; a, b, c

2.1 Recognising Quadratic Equations

Exercise Module Q.E page1

2.3 To Solve Quadratic Equations

Students will be taught to 2. Understand the concept of quadratic equations.

Students will be able to: 2.1 Determine the roots of a quadratic equation by ( a ) Factorisation ( b ) completing the square ( c ) using the formula

Method 1 By Factorisation This method can only be used if the quadratic expression can be factorised completely.

Solve the quadratic equation x 2  5 x  6  0

Answer :

x2  5x  6  0

 x  2   x  3  0 x  2  0 or x  3  0

x  2 or x  3

Method 2 Formula x 

b 

b 2  4 ac 2a

Solve the quadratic equation 2 x 2  8 x  7  0 answer correct to 4 significant figures

by formula.Give your

Answer :

b  b  4ac x 2a 2

a=2 , b =-8, c=7

 (8)  (8) 2  4(2)(7) x 2(2)

8 8 x 4 x = 2.707 atau 1.293

Method 3 By Completing The Square - To express ax  bx  c  0 in the form of a  x  p   q 2

2

Simple Case : When a = 1

2 Solve x  4 x  5  0 by method of completing square

x  4x  5  0 

2

2

4  4    5  0 2  2

x2  4x   

x  4x   2   2  5  0 2

2

 x  2

2

45  0

 x  2  9  0 2  x  2  9  x  2  9 2

2

2

x  2  3 x  3 2

x  3  2

x 1

x  5

Method 3 By Completing The Square - To express ax  bx  c  0 in the form of a  x  p   q 2

2

[a = 1, but involving fractions when completing the square]

2 Solve x  3 x  2  0 by method of completing square

 x  2

x 2  3x  2  0 2

2

 3  3 x  3x        2  0  2  2 2



2



 x

3  9     2  0 2  4



2

x  2

17 0 4 17  4



2

x2 

17 4

17 x  2 4 x = 3.562 or

x 

x = - 0.5616

17 2 4

Method 3 By Completing Square - To express ax  bx  c  0 in the form of a  x  p   q 2

2

If a ≠ 1 : Divide both sides by a first before you proceed with the process of ‘completing the square’.

2 Solve 2 x  8 x  7  0 by method of completing square

2 x2  8x  7  0

2

2 x2 8x 7 0    2 2 2 2 7 x2  4x   0 2 2

first 

2

4

 x  2

2



 x  2

7 0 2

1 0 2



1 2

2

7  4  4 x  4x         0 2  2  2 2

 x  2

2.707 or 1.293

1. Solve quadratic equation x 2  4 x  5  0 by factorisation. 2. Solve quadratic equation 2 x( x  1)  6 by method of completing the square 3. By using formula,solve quadratic equation ( x  1) 2  1

Module Q.E page 4

2.4 To Form Quadratic Equations from Given Roots

Students will be taught to 2. Understand the concept of quadratic equations.

Students will be able to: 2.2 Form a quadratic equation from given roots.

2.4 To Form Quadratic Equations from Given Roots If the roots of a quadratic equation are α and β, That is, x = α , x = β ; Then x – α = 0 or x – β = 0 , (x – α) ( x – β ) = 0 The quadratic equation is

x 2  (   ) x    0

x 2  Sum of roots x

product of roots

0

x 2  ( sum of roots ) x  (Pr oduct of roots )  0 Find the quadratic equation with roots 2 dan- 4.

x=2,x=-4

x 2  (   ) x    0

SOR  2  (4)  2

x 2  (2) x  8  0

POR  (2)(4)  8

x2  2 x  8  0

2.4 To Form Quadratic Equations from Given Roots Given that the roots of the quadratic equation 2 x 2  ( p  1) x  q  2  0 are -3 and ½ . Find the value of p and q.

1 x  3, x  2 SOR  3 

1 5  2 2

1 3 POR  ( 3)( )  2 2

5 3 x  x 0 2 2 2

2 x  5x  3  0 2

Compare 2 2 2 x  ( p  1) x  q  2  0 2 x  5 x  3  0 and

x 2  (   ) x    0

p 1  5

q2 3

5 3 x  ( )x   0 2 2

p4

q5

2

L1. Find the quadratic equation with roots -3 dan 5. L2. Find the quadratic equation with roots 2 dan- 4.

Module page 9

2.5.1 Relationship between

b 2  4ac

and the roots of Q.E

Students will be taught to 3. Understand and use the condition for quadratic equations to have ( a ) two different roots ( b ) two equal roots ( c ) no roots

Students will be able to: 3.1 Determine types of roots of quadratic equation 2 from the value of .b  4ac

2.5 The Quadratic Equation 2.5.1 Relationship between

ax 2  bx  c  0

b 2  4ac

and the roots of Q.E

Case 1 b 2  4ac  0 Q.E. has two distinct/different /real roots. The Graph y = f(x) cuts the x-axis at TWO distinct points.

2.5 The Quadratic Equation 2.5.1 Relationship between

ax 2  bx  c  0

b 2  4ac

and the roots of Q.E

Case 2 b 2  4ac  0 Q.E. has real and equal roots. The graph y = f(x) touches the x-axis [ The x-axis is the tangent to the curve]

2.5 The Quadratic Equation 2.5.1 Relationship between

ax 2  bx  c  0

b 2  4ac

and the roots of Q.E

Case 1 b 2  4ac  0 Q.E. does not have real roots. Graph y = f(x) does not touch x-axis.

Graph is above the x-axis since f(x) is always positive.

Graph is below the x-axis since f(x) is always negative.

2 The roots of quadratic equation 2 x  px  q  0

are -6 and 3

Find (a) p and q, 2 (b) range of values of k such that 2x  px  q  k does not have real roots. ( a) x = -6 and x=3

2 x 2  6 x  36  k

( x+6 )( x-3 )=0

2 x 2  6 x  36  k  0

x 2  3 x  18  0

a=2

2x 2  6 x  36  0 Comparing

c=-36-k

does not have real roots.

b 2  4ac  0

2x 2  px  q  k P=6

b= 6

q = -36

62  4(2)(36  k )  0

324  k 8  0 k  40.5

2 2x xk 1. Find the range of k if the quadratic equation

has real and distinct roots. 2. Find the range of p if the quadratic equation 2 x 2  4 x  p  0 has real roots.

Module page 9