PERFECT SQUARES 1

What must be added to x²  + 12x to make the expression a perfect square?

Solution

a=1, b=12,c=?

b² = 4ac

(12)² = 4 x 1 x c

144 = 4c

144 = 4c

 4       4

36 = c

Hence number to be added is 36

TRY THIS……………………

What must be added to x²  + 16x to make the expression a perfect square?  

LOGARITHMS 19

Change the following into Logarithmic form.

i)  4³ = 64

ii)  2³ = 8

iii)  7⁰ = 1

Solution

4³ = 64 ⇔ log₄ 64 = 3

2³ = 8 ⇔ log₂ 8 = 3

7⁰ = 1 ⇔ log₇ 1 = 0

TRY THIS………………….

Change the following into Logarithmic form.

i)  3² = 9

ii)  2⁴ = 16

iii)  11⁰ = 1 

VARIATION 2

x is directly proportional to y. x=36 while y=4. Find y when x is 720.

Solution

x ⍺ y

x = ky

36 = k x 4

36 = 4k    [dividing by 4 both sides]

4      4

k = 9

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

k = 9, y= ? x = 720.

x = ky

720 = 9 x y

720 = 9y

9       9

y = 80.

TRY THIS…………….

x is directly proportional to y. x=40 while y=5. Find y when x is 496.

LOGARITHMS 18

Simplify Log₂1024 - Log₃27

solution

= Log₂1024 - Log₃27

= Log₂2¹⁰ - Log₃3³

= 10Log₂2 - 3Log₃3

= (10 x 1) - (3 x 1)

= 10 - 3

= 7

Hence Log₂1024 - Log₃27 = 7

TRY THIS..................

Simplify Log₂2048 – Log₅625

ALGEBRA 9

Multiply 4x - 7y by -5a

solution

= -5a(4x - 7y)

= (-5a x 4x) + (-5a x -7y)

= -20ax + 35ay     [since -5a x -7y = +35ay ]

= -20ax + 35ay    answer

TRY THIS...................

Multiply 8p - 3q by  -6a

FUNCTIONS 5

If f(x) = x⁴ + kx² + 6x + 7 has a remainder of 24 when divided by x+2; find k.

solution

f(x) = x⁴ + kx² + 6x + 7

x + 2 = 0

x = -2

f(x) = (-2)⁴ + k(-2)² + 6(-2) + 7 = 24

16 + 4k + (-12) + 7 = 24

16 + 4k - 12 + 7 = 24

16 + 4k - 5 = 24

4k + 16 - 5 = 24

4k + 11= 24

4k = 24 - 11

4k = 13

4k = 13

4      4

k = ¹³/4

hence k=¹³/4

TRY THIS......................

If f(x) = x⁴ - kx² + 3x - 11 has a remainder of 12 when divided by x+3; find k.

ALGEBRA 8

Divide 78ax + 6ay - 10az by 2a.

Solution

= 78ax + 6ay - 10az

               2a

= ³⁹78ax  +  ³6ay  -  ⁵10az       [cancelling by 2a throughout]

      2a           2a          2a

= 39x + 3y - 5z

TRY THIS...................

Divide 28ae - 120av + 64aw by 4a

RECIPROCAL-4

Find the reciprocal of m¹³

Solution

=   1      

     m¹³

= 1 x ¹/m¹³

         

= 1 x m^-13     [since ¹/aⁿ = a^-n]

= m^-13

TRY THIS........

Find the reciprocal of w⁸⁴

FACTORIZING 10

Factorize 2x² - x - 15

Solution

= 2x² - x – 15. 

We split the middle term (-x)

We look for two items such that

mxn =  2x² x (-15)= -30x²and

m+n = -1x

The correct choice is -6x and 5x:

= 2x² - 6x + 5x – 15            Since    (-x) = -6x + 5x.

= (2x² -6x) + (5x - 15)    grouping the factors

= 2x(x - 3) + 5(x - 3)  factoring out common terms

= (2x + 5) (x - 3)  answer

Hence 2x² - x – 15 = (2x + 5) (x - 3) answer.

TRY THIS…..

Factorize 7x² - 27x – 4.

LOGARITHMS 17

Evaluate Log₂(256 ÷ 8).

Solution

= Log₂(256 ÷ 8)

= Log₂256 -  Log₂8          (applying the product rule)

= Log₂2⁸ -  Log₂2³             ( 256= 2⁸ and 8=2³ )

= 8Log₂2 -  3Log₂2       ( remember  Log_aa^c = cLog_aa )

= (8 x 1) -  (3 x 1)          ( remember  Log_aa = 1 )

= 8 - 3

= 5 answer

TRY THIS...............

Evaluate Log₂(1024 - 32).