Lesson Notes By Weeks and Term - Senior Secondary School 1

SUBJECT: MATHEMATICS

CLASS:  SS 1

DATE:

TERM: 2nd TERM

REFERENCE BOOK

• New General Mathematics SSS 1 M.F. Macrae et al 
• WABP Essential Mathematics For Senior Secondary Schools 1 A.J.S Oluwasanmi

 
WEEK TWO

TOPIC: General form of quadratic equation leading to Formular method

CONTENT

• Derivative of the Roots of the General Formof Quadratic Equation.
• Using the FormularMethods to solve Quadratic Equations
• Sum and Product of quadratic roots.

Derivative of the Roots of the General Form of Quadratic Equation

The general form of a quadratic equation is ax2 + bx + C = 0. The roots of the general equation are found by completing the square.

ax2 + bx + C = 0

Divide through by the coefficient of x2.

ax2  +bx +  C   =  0

aaa

x2 + bx + C    = 0

aa

x2  +  b   x    = 0  -   C

aa

x2 + bx  =  - C

aa

The square of half of the coefficient of x is 

        ½ x b 2   =      b 2

a                2a

Add       b2  to both sides of the equation.

 2a

x2 + bx +        b 2= - C+b2

2a2aa    2a

x+b2a2= - C+ b2

a       4a2

           x +  b2   =  - 4ac  + b2

                  2a                      4a2

i.e       x +b 2   =      b2 – 4ac

                2a                     4a2

Take square roots of both sides of the equation :

    x+b2a2 = b2-4ac4a2

i.e  x + b= ± √ b2 – 4ac

2a    2a

x =   -b2ab2-4ac4a2

Hence 

    x  =  -b ±√ b2 – 4ac

                             2a

EVALUATION

Suppose thegeneral  quadratic equation is Dy2 + Ey  + F = 0

Using the method of completing the square, derive the roots of this equation

Using the FormularMethods to Solve Quadratic Equations

Examples

Use the formula method to solve the following equations. Give the roots correct to 2 decimal places:

1. 3x2  - 5x – 3 = 0
2. 6x2 + 13x + 6 = 0
3. 3x2 – 12x + 10 = 0

Solution 

1. 3x2 – 5x – 3 = 0

Comparing 3x2 – 5x – 3 = 0

With ax2 + bx+  C = 0

a = 3, b = -5, C = -3

Since 

X = -b ±√b2 – 4ac

                      2a

x = -(-5) ±√ (-5)2 – 4 x 3 x -3

2 x 3

x =  + 5  ± √ 25 + 36

                          6

x =    + 5 ±√61

               6

x = + 5 + 7.810    =  + 12.810

                 6                    6

or

x = +5 – 7.810     = -2.810

               6                   6

  x =  12.810 or x =  - 2.810

             6                           6

x =  12. 810     or x =- 2.810

              2                       6

i.e.x = 2.135  or x = -0.468

x = 2.14 or x = -0.47

to 2 decimal places

(2) 6x2  + 13x  + 6=0

comparing 6x2  + 13x  + 6=0

with ax2 +bx + c = 0

a= 6, b =13,  c = 6

Since 

x =-b ± √ b2 – 4ac

                      2a

x = - 13 ±√ (13)2 – 4 x 6 x 6

                             2 x 6

x = -13 ±√169  - 144

                         12

x =- 13 ±√25

           12

x = =-13 ± 5

            12

x =  -13 + 5  or x = -13 – 5

           12                     12

x = -8    or x = - 18

        12               12

x= -2    or x = -3

1. 2

x=- 0.666 or x = - 1.50

i.e x= 0.67 or x = -150 to 2 decimal places .

(3)  3x2 – 12x + 10 = 0

comparing 3x2 – 12x + 10 = 0 with ax2 +bx + c = 0, then 

a = 3, b= -12, c = 10.

Since 

X = -b ± √b2 – 4ac 

          2a

then

x = - (-12) ±√(-12)2 -4 x 3 x 10

                              2 x 3

x = + 12 ±√ 144 – 120

                 6

x =  + 12 ±√24

     6

x =  12 ± 4.899

                6

x =   + 12  + 4.899   =  16.899

                   6                     6

or x =  + 12 – 4.899   =   7.101

1. 6

i.e x  =   16.899   or x =  7.101

                   6                       6

x = 2.8165  or x = 1.1835

i.e . x = 2.82 or x = 1.18 to 2 decimal places.

EVALUATION 

Use the formula method to solve the  following quadratic equations .

1. t2 – 8t  + 2 = 0
2. t2 + 3t + 1 = 0

• Sum and Product of quadratic roots.

We can find the sum and product of the roots directly from the coefficient in the equation

It is usual to call the roots of the equation α and β  If the equation 

    ax2 +bx + c = 0   …………….  I 

has the roots α and β then it is equivalent to the equation

    (x – α  )( x – β )  = 0

x2-∝+βx + ∝β=0 ………… 2

Divide equation (1)by the coefficient  of x2

 ax2+ bx + c   = 0  …………   3

aaa

Comparing equations (2) and (3)

x2 +  b x  +  c  = 0

aa

x2  -  ( α +β)x  + αβ    = 0

then

α+ β= -b

a

and αβ = c

a

For any quadratic equation, ax2 +bx + c = 0 with roots α and β

α + β  = -b

a

αβ =  C

a

Examples

1. If the roots of 3x2 – 4x – 1 = 0 are αand β, find  α + β and αβ

2. If α and βare the roots of the equation 

     3x2 – 4x – 1 = 0 , find the value of 

(a)     α     +  β

β          α

(b)   α   -   β

Solutions

1a.  Since α + β =  -b

a

Comparing the given equation 3x2 – 4x – 1= 0 with the general form

ax2 + bx + c = 0

a = 3, b = -4, c = 1.

Then 

α + β = -b  =-(-4)

a         3

        = + 4  =  +1 1/3

               3

αβ =c  =  -1 = -1

a        3     3

2. (a)α + β =  α2 +β2

β      α          αβ

= (α + β )2 - 2αβ

            αβ

Here, comparing the given equation, with the general equation,

a = 3, b = -4, c = - 1

from the solution of example 1 (since the given equation are the same ),

α + β = -b =  - (-4)   = +4

a           3           3

αβ  = c = - 1

a       3

then

α  +    β =  ( α+ β ) 2 – 2 αβ

β        α                  αβ

=  (4/3 ).2 – 2 (- 1/3)

        -1/3

=   16    + 2

       9      3

- 1

            3

=   16 + 6   ÷    -1

         9        3

22  x   -3

       9          1

=   -22

       3

or  α + β     = - 22   = - 7 1/3

β        α           3

(b)  Since 

(α-β) 2 =α2 + β 2-2α β

but

α2 + β2 = (α + β)2 -2 α β

:.(α- β)2 = (α+ β)2 - 2αβ -2αβ

(α – β)2 = (α + β)2 - 4α β

:.(α – β) = √(α + β )2 - 4αβ

(α – β) =√(4/3)2 – 4 (- 1/3)

= √16/9  +4/3

 =    16+129

 = 289          =   283

:. α - β  =   √28 

3

EVALUATION 

If α and β are the roots of the equation 2x2 – 11x + 5 = 0, find the value of 

1. α - β
2. 1∝+1+1β+1

GENERAL EVALUATION 

Solve the following quadratic equations:

1. 63z = 49 + 18z2
2. 8s2 + 14s = 15 

Solve the following using formula method:

1. 12y2 + y – 35 = 0
2. h2 – 15h + 54 = 0 

READING ASSIGNMENT 

New General Mathematics SS Bk2 pages 41-42 ,Ex 3e Nos 19 and 20 page 42.

WEEKEND ASSIGNMENT 

If α and β are the roots of the equation 2x2 – 7x – 3 = 0 find the value of:

1. α  + β   (a) 2/3    (b) 7/2    (c) 2/5    (d) 5/3
2. α  β  (a) -3/2   (b) 2/3     (c) 3/2   (d) – 2/3
3. α β2  + α2 β   (a) 21/4    (b) 4/21   (c) – 4/21   (d) -21/4

Solve the following equation using the formula method.

1. 6p2 – 2p – 7 = 0
2. 3 = 8q – 2q2.

THEORY

1. Solve the equation 2x2 + 6x + 1 = 0 using the formula method
2. If α and β are the roots of the equation  3x2 -9x + 2 = 0, find the values of 

1. α β2  + α2β
2. α2 - αβ +  β2