ALGEBRA 2

If 13w² = 325; find w

Solution

13w² = 325

13w² = 325               [Dividing by 13 both sides]

13          13

w² = 25

w = 5          Because the square root of 25 is 5.   

Hence w = 5 .          

TRY THIS……………………….

If 14y² = 5600; find y.

POLYGONS 2

A regular polygon has 27 sides. Find the total interior angles of that polygon.

Solution

n = 27

Total angles = (n - 2)180⁰

                   = (27 – 2)180⁰

                   = 25 x 180⁰

                   = 4500⁰

Total angles = 4500⁰

TRY THIS……………………….

A regular polygon has 38 sides. Find the total interior angles of that polygon.

EXPONENTIALS 2

If 8¹⁰ = 32^a-2; find a

Solution

8¹⁰ = 32^a-2 

(2³)¹⁰ = (2⁵)^a-2 

2³⁰ = 2^5(a-2) 

2³⁰ = 2^5a-10

2³⁰ = 2^5a-10  [Same bases both sides cancels out]

30 = 5a – 10

30 + 10 = 5a

40 = 5a

40 = 5a

5      5

8 = a

Hence a=8

TRY THIS……………………….

If 5¹⁰ = 625^h-2; find h

QUADRATICS 1

What must be added to x²  + 20x to make the expression a perfect square?

Solution

a=1, b=20,c=?

b² = 4ac

(20)² = 4 x 1 x c

400 = 4c

400 = 4c

4       4

100 = c

Hence number to be added is 100

TRY THIS……………………

What must be added to x²  + 10x to make the expression a perfect square?

FACTORIZE 1

Factorize completely mc + mr - cr - c².  

Solution

= mc + mr - cr - c².

= (mc + mr) – (cr - c²).      [Grouping the factors]

= m(c + r) - c(r + c).   [after factoring out]

 = (c + r)(m - c) answer

  Hence mc + mr - cr - c². =  (c + r)(m - c).   

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

Factorize completely pw + pr - rw - w².

UNITS OF LENGTH 1

Convert 53734m into km.

Solution

1km  = 1000m

  ?     = 53734m

After cross-multiplying;

= 1 x 53734

      1000

=   53734

     1000

= 53.734 km

Hence 53734m = 53.734km

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

Convert 72682m into km.

VARIATIONS 1

x is directly proportional to y. x=32 while y=4. Find y when x is 736.

Solution

x ⍺ y

x = ky

32 = k x 4

32 = 4k    [dividing by 4 both sides]

4      4

k = 8

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

k = 8, y= ? x = 736.

x = ky

736 = 8 x y

736 = 8y

8       8

y = 92.

TRY THIS…………….

x is directly proportional to y. x=40 while y=4. Find y when x is 380.

APPROXIMATIONS 1

Estimate the value of 2.1  x  0.034

solution

= 2.1  x  0.034

= 2.0 x 0.03    [ 2.1 ≈ 2.0 to ones and 0.034 ≈ 0.03 to hundredths]

= 0.06 

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

Estimate the value of 5.4  x  0.073

LOGARITHMS 2

If log₃(10x - 23)=3; find x.

Solution

log₃(10x - 23)=3

 (10x - 23)=3³       

                                            

 10x - 23=27

10x = 27 + 23

10x =  50

10x =  50⁵   

10      10

x = 5

Hence x = 5

TRY THIS…………….

If log₃(5x + 6)=4; find x. 

FUNCTIONS 1

Find a linear function f(x) with gradient -6 which is such that f(5)=14.

Solution.

m=-6, points = [5,14] and [x, f(x)]

m = y₂-y₁/x₂-x₁

-6 = [f(x) – 14]/x-5

f(x)-14=-6(x-5)  [after cross multiplying]

f(x)-14=-6x+30

f(x)=-6x+30+14

f(x)=-6x+44

A  linear function is f(x) = -6x + 44.

TRY THIS……….

Give out a linear function f(x) with gradient -4 and f(5)=11.