CPhill

 x - 2               10x

_____    +    _______

x + 3              x^2  - 9

x - 2                  10x

____    +      _________

x + 3            (x + 3) (x - 3)

(x - 2) (x - 3)                10x

___________   +     ____________

(x + 3) ( x - 3)           (x + 3) ( x - 3)

[ x^2 - 5x + 6 + 10x ]

_________________

(x + 3) ( x - 3)

[ x^2 + 5x + 6 ]

_____________

{ x + 3] [ x - 3]

[ x + 2] [ x + 3]

____________

[x - 3] [ x + 3]

x + 2

____                 restriction :   x not equal to  -3 or 3

x  - 3

Angle C  = 60°

And the side opposite this  is  √3  times as long as the side opposite the 30° angle

So AB  =  6√3

So......the area is

(1/2) height * base

(1/2) AC * AB  =

(1/2) * 6 * 6√3  =

18√3  units^2

One way for triangles to be similar is that we need that the ratios of the corresponding sides to be =

That is

short side of first triangle

_____________________        =  A

short side of second triangle

Intermediate side of first triangle

__________________________        =  B

Intermediate side of second triangle

Long side of first triangle

_______________________     =   C

Long side of second triangle 

So  A = B  = C   for the triangles to be similar

So.....For 16 to be similar...we need that

23/40  = 34/55  = 36/60

23/40 = 34/55 = 3/5

None of these ratios are equal...so....not similar

For 17  to be similar....we need the sides around the equal angle need to be equal ratios

But

16/22   =  25/33  ???

8/11 = 25/33     these are not equal

For the same reason....the triangles in 19 aren't similar, either

In other words  14/9  not equal to 7/4

And the triangles in 20 are not similar because  we can't establish any AA similarity

√-1   is defined as  i

And  i^2  =  -1

So....we have

(i)^15  =

(i^2)^7  * i  =

(-1)^7  * i  =

-1 * i  =

-i

x^3 - x^2 + x - 1  factors as

x^2 (x -1) + 1(x - 1)  =

(x - 1)(x^2 + 1)

x^2  - 1   factors as

(x + 1) ( x - 1)

Taking all the different factors we have

(x - 1)  (x + 1) (x^2 + 1)   ⇒  second answer

  7                     7

____    +       ______

a + 8              a^2  - 64

  7                       7

_____   +     __________

 a + 8           (a + 8) (a - 8)

    7(a - 8)                         7

__________     +         _________

(a + 8) (a - 8)              (a + 8) (a - 8)

[ 7a - 56  + 7 ]

____________

(a + 8) (a - 8)

[ 7a - 49 ]

___________

(a + 8) (a - 8)

Thanks, EP  !!!

15...if these are similar....we must have that

13/15  =  30/35

13/15  = 6/7      not true....so these are not similar

16.....if these were similar.....we must have that

40/60  = 23/34

2/3  =  23/34  ....not true.....so.....not similar

17  For these to be similar...we must have that

16/22  = 25 /33

8/11  = 26/33      .....not true....not similar

18 EFG  similar to EBC

Angle E  = Angle E  and

EF / EG  = EB/EC

50/60  = 20/24

5/6  = 5/6

19 Not similar

20 Not similar

11 - correct

12 - correct

13...if these are similar....we must have that

54 / 60   =  62 / 70

9 / 10  =   31 / 35         not true....so....these are not similar

14  - correct

36 is correct  !!!!

y = x^2 -3x + 4      (1)

x + y  = 4  ⇒   y  = 4 - x    (2)

Sub (2)  into (1)

4 - x  =  x^2  - 3x + 4       subtract 4 from both sides

-x = x^2 - 3x       add x to both sides

x^2 - 2x  = 0   factor

x( x - 2)  =  0

Set each factor to 0 and solve for x and we get that

x =  0   or x  = 2

When x = 0.....y  =  4 - 0  =    4      ⇒   (0, 4)

When x  =  2.....y  =  4 -  2   =  2  ⇒    (2 , 2)