ROUTINE INFORMATION Hillview Secondary School
Name of School: Student Surname and Name:
Maphumulo Mthokozisi
Student Number:
207503878
Grade:
10, 11 and 12.
Subject:
Mathematics
Topic: Content /Concept Area:
Quadratic Equation –the completion of square and long division. Completion of square and long division.
CAPS page number:
193
Duration of Lesson:
45 minutes
Date
July / August 2018
Aims: To develop learners skills for being able to deal with quadratic equation involving ax2 + bx + c = 0, where a, b and c are real numbers and a ≠ 0. To develop learner’s skills for being able to solve equations by factorising sticking to rules of how we factorise, divide and complete the equations related to quadratic equations. To make learners develop capabilities for being able to inspect and diagnose discriminant for general quadratic equation used to determine the nature of roots whether they are having D =0, D<0, D>0 , where D = b2−4ac. Develop learner’s skills for being able to s: solve equations, completing the square and apply quadratic formula when factorising and doing long division involving binomials, polynomials and even calculus for differentiation and curve graphs.
Lesson Objectives:
Knowledge
Learners should gain
Skills Learners should be able to do
Value /Attitude Learners should acquire
knowledge and
following :
values and attitudes
understanding of the
conducive to :
following : Define, determine and analyse
Demonstrate how co-efficient 2
Acknowledge
ability
to
the difference of two (2)
of x and constant term fits to
compete the square of the
squares and factorise.
the conversion of quadratic
quadratic equation given by
Describe where, when and how
equation. Determine the nature of the
f(x) = x2 + bx + c. Acknowledge ability to be able
the quadratic equation become
roots in the quadratic equation.
to examine roots of quadratic
useful. Hint use Discriminant,
Examine Discriminant (D or
equation.
difference of two squares to
∆¿.
reveal roots (nature of roots in
b −4 ac
Where
D
=
2
the quadratic equations). Optional : Give a brief
Optional Acknowledge ability
overview of application of
Optional: Determine the axis
to explore conjectures related
quadratic equation and its
of symmetry, turning point and
to completion of square on the
square completion in the eother
y – intercept of a function of a
quadratic equation and apply
functions such as parabola
parabola.
their
knowledge
binomials,
into
trinomials
polynomial
and
algebraic
expressions. (Hence integrate understanding
of
quadratic
function into long division and system used to draw, analyse and
determine
points
of
parabola) on which y = a(x-p) 2 + q. which is similarly to both y = ax2 + bx +c and y = x 2 + q . (Either way round).
Approaches / Teaching Strategies:
Role plays. Case studies Class discussions. Questioning. Whole class-discussion Group Work
Resources: Maps , orthophoto and topographic map) Calculators Chalkboard Newspaper Worksheet
During the introduction question learners to explain how they go about solving equation involving the difference of twoo squares. In doing this we are making them to observe how does a binomial expression is trawnsformed to take roots leaving it form of y = ax 2 + bx + c. Learners must explain how they move from x2- 16 to (x-4) ( x+4) or from (x 2-3x + 9) to ( x-3) (x-3). Therefore this serve as a point to make learners recall factors of common algebraic expressioons. In fact learners are expected to describe quadratic equation with respect to ax 2 + bx + c , in which a is a co-efficient of x 2 , b is cooffiecient of x and c is a point in which a graph of y = ax 2 + bx + c â&#x20AC;&#x201C; crosses the y- axis. Other books call c â&#x20AC;&#x201C; as y-inteercpt. And then the entire equation of f(x) = y = ax 2 + bx + c is a an expression of a parabola. In a straight line graph os a linear function where y = mx+ b , b is till representing c , because it is a y-intercept.
Development : It is time to demosntrate quadratic equation of y = ax2 + bx + c. Learners must be able to identify the value of a, b, c across different expressions that appear as quadratic equations. For that reason they
must begin to learn how to complete a aquare of a quadratic equations . Following then few examples are given as their exercise . Thearfter more advanced applications of quadratic equations are going to be given. Complete a square of quadratic equation. Step 1: Generally y = ax2 + bx + c. Let y= f(x) = 0
∴ ax2 + bx + c = 0 Step 2 divide the entire expression by 2
ax 2 a
bx a
+
x2+
bx a
+
x2 +
bx a
=-
+ c a
c a
=0
=0 c a
Step 3: to complete the auare of x 2+
bx , ou must half the centre of y = ax 2 + bx + c and a
then multiply the entire combination as (
1 2
x
b x ) and then apply squaring of the a
complementary root to the entire equation. This means that you apply (
1 )2 both sides . 2a
The rules says you ignore any number that may appear with a quadratic equation as long as you would be able to apply a square from both sides of an equation ( quadratic equation ). But one can ask ,what is ging to happen with the root c , which has alwas been there . Th answer is simpler , you need to simple push it to the RHS of an equation . It up to you when , how you push it . NB when you half or apply half to the b – the coefficient of x – you need to know that it is quoted withthout intact / directly contacting the sets st centre . So you shoukd make quotes
aside and then complete a square as (1/2 of B) 2 full stop . You apply the (1/2 of B) 2 to both sides of equation , without having to touch the value of c or b in real equation . In other words you are dealing with different kind of factorisation where you must be able to move from polynomial expression to difference of two squares.
x2 +
bx a
c
0
+ a = a
x2 +
bx a
+ a =0
c
x2 +
bx a
=- a
c
bx a
We then complete the square for x2 +
by adding (
b )2 2a b
This is take ½ of a x all tossed into [ B ]
2
, where B=
b
½ of a x excluding x ∴
x2 +
bx b +( ) 2 a 2a
c
b
= - a + ( 2 a )2
b
b
b
c
( X + 2 a ) ( X + 2 a ) = ( 2 a )2 - a b
b
b
c
( X + 2 a )2 = ( 2 a )( 2 a ) - a b
b2 4 a2
b
b2−4 ac 4 a2
b
b −4 ac = ±√ 2
( X + 2 a )2 = ( X + 2 a )2 =
c
- a
2
X + 2a
4a
2
= ±√ b
X + 2a X=
b
- 2a
b −4 ac 2 a2 b
= - 2a
±√
b2−4 ac 2a
−b ± √ b2−4 ac 2a
NB what ever you do with B from LHS you also Do it across the RHS .
b x and get it all tossed a into [ B ] 2
Take a ½ of
Where B = ½ of In other word let B =
b x a b x 2a
excluding x. Remember you do that with discretion to the values of
b x ... so you will a
prepare the value of B aside a, which is not different with how you deal with completion of difference of two squares. Correct and accurate completion of square will rely upon how you tossed
b x into [ B ] 2. The word a
½ of
tossed is ‘substitution’ at inverted commas. Never forget to live x2 +
bx a
bx ¿ a
without an intact*.
NB we need to transform x2 +
bx a
into x2-y2. You perfomr
square completion with descretion to the coefficient of x , without having to touche x2 +
bx . a
So factorise outside , transpose remaining values using addiidctive and subractive inverses . ***
Classroom exercise : Given a quadratic equation is x2 + 5x + 3.
a) Using application of knowledge earned from square completion , prove that the copletion of square of f(x) is given **by (x 2 + (hnt let y = f(x) = 0) b) Exemine a discrimant of equation of f(x) = x 2
+
5 2
13
x)2 - 4
.
5x + 3 also using
knowledge of quadratic eqiation given by y = ax2 + bx + c. ( hint ∆ = ). c) Comment on the roots of the equation given above . d) Where do you think this completion may apply ? ( hint : y = a( x-p)2 + q). Challenges e) Is quadratic equation likely to lend itself to long division , if yes … b2−4 ac
explain in which way ( hint f(x) = x 2 + 5x + 3. ( quotent and reminder theorem ) Solution : Given f(x) = x2 + 5x + 3. Let y = f(x) = 0 ∴ 0=¿ x2 + 5x + 3. x2 + 5x +3 = 0 change LHS int RHS and vice versa. For f(x ) = ax2 + bx + c. in the above euation , a = 1, b = 5 and c = 3. Rule is take
1 2
b x excluding x – the 2a
of bx and substitute into (B) 2, where B = B =
constant term. This simple means B =
b x with descrition from ax2 + bx 2a
Threfore x2 + 5x +3 = 0 Similarly x2 + 5x + (
5 ¿ 2
1 b of = 2 2a 5 5 of = 2 2( 1) B=
2
+3 = 0 + (
5 ¿ 2
2
∴ (B)2 = (
5 ¿ 2
2
=
The next step is to factories x2 + 5x, which is to complete a square of ax2 + bx. You have a mirror of x2-y2 which is a difference of two (2) squares where (x-y) ( x+y) .
1 2
Same thing goes with x2 – 3x + 9, where x2 – 3x + 9 = (x-3) (x-3) = (x-3)2 . Similarly : x2 + 5x + (
(x+
5 )(x+ 2
5 ¿ 2
2
+3 = 0 + (
5 5 ¿ ) +3 =( 2 2
2
5 ¿ 2
2
. Wharever you do to the LHS with B you also do it
to the RHS. (x+
(x+
5 2 5 ¿ ) =( 2 2 5 2 ) = 2
25 4
2
- 3 (hint addictive, subtractive inverse). –3
(x+
5 2 ) = 2
25 4
(x+
5 2 ) = 2
25−12 4
(x+
5 2 ) = 2
13 4
(x+
5 2 ) 2
13 4
–
3 1 ( find LCD and complete the fraction ).
=0
That is why we arge that x2 + 5x + 3 = (x2 + 5
13
**by (x2 + 2 x)2 - 4
5 2
13
x)2 - 4
.
NB Wharever you do to the LHS with B you also do it to the RHS and vice versa.
Assessement: Textbooks exmples are given. ( feedback will follow after their completion of task , exemining how they applied their independent knowledge – through sharing ideas and developing their own ideas).
Solution :Planned questions :
1) 2) 3) 4)
What is difference of two squares ( hint x2 – y2). What does completion of square in the quadratic equation entails. What is addictive inverse and what is subtractive inverse – illustrate Describe where could one expect to use quadratic equations , where posible draw a
function using geogebra or grid pattern 5) What is discrimination and how does it work ?
Consolidation and Conclusion : For most people , they felt quadratic equatiom is used to solve equation of a bionomial whose roots cannot be factorised completely. The converse is true , however the square completion has muptipurpose application. You can use completion of square for devicing a mthod used to determine the nature of roots where discriminant is given by discriminant of D = b2 – 4ac. This is often use for determining nature of roots the quadratic equation has. Note: b 2 – 4ac comes from the quadratic formula.