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For the equation to be undefined, the denominator must equal 0, because anything divided by 0 is undefined. 

Thus, we have the equation: \(y^2-4y+4=0\), where we have to solve for y.

To solve for y, use the quadratic formula: \(y = {-b \pm \sqrt{b^2-4ac} \over 2a}\) , because the quadratic is in the form \(ax^2+bx+c=0\)

Can you do it from here?

You will never be able to win 90%  of your games  because  you are only winning 2/3 = 67%  of the games you play after the 5th week......you would have to win > 90% of the games you play after the 5th week to  make this possible

Thank you!

The slope of  the line perpendicular to the given line  is   -2

A line with this slope going through A  has the equation

y  = -2 ( x -3) + 1

y = -2x + 7

Setting the two equations   equal  to  find the x intersection of  these lines we have

(1/2)x + 2 =  -2x + 7

(5/2)x  =  5

x =  5 ( 2/5)   = 2

And the y intersection = -2(2) + 7   =   3

So the intersection pt  is (2,3)

Using the midpoint rule we can find the coordinates of B as

[ 3 + x] / 2 = 2   ⇒    3+ x = 4   ⇒    x  =1

[1 + y] / 2  =  3   ⇒ 1 + y = 6 ⇒   y =  5

So

B = ( 1 , 5)

In this problem, any x value that makes the denominator =  0   will produce an asymptote at that x value

We can write the denominator as

(x - 1) ( x - 2)   =

x^2  - 3x  +  2

a  = -3      b =  2        their sum is  -1

The diagonal length  =   20sqrt 2  cm

[The diagonal of a sqaure is  always   side length * sqrt 2 ]

x^2  +  y^2  =  4      (1)

(x - 5)^2  + (y - 5)^2 = 40   ⇒   x^2 + y^2  - 10x - 10y + 25 + 25 =  40  ⇒

4 - 10x - 10y  + 50  =  40 ⇒   -10x -10y =  -14  ⇒    5x + 5y  = 7  ⇒   5y = 7 -5x ⇒  y = (7 -5x) / 5    (2)

Sub (2)  into (1) for y

x^2 +  [ (7- 5x)/5)^2  = 4

x^2  +  [ 49 - 70x + 25x^2 ] / 25   =  4          multiply through by 25

25x^2  + 25x^2 - 70x  + 49   = 100

50x^2 - 70x - 51  =   0

The approx solutions to  this  are    x ≈ -.528  and  x ≈ 1.92

And  when   x ≈ -.528   y ≈1.92

And when x  ≈ 1.92   y ≈ -.528

The slope  is   [   1.92  - - .528 ] / [ -.528 - 1.92]   =  [1.92 + .528 ] /  - [ 1.92 + .528 ]  =   -1

The number of 10-digit numbers where the sum of the digits is divisible by 10 is 2,000,000.

We can expect   five  composite rolls for  every prime  roll....so....

5 * 185 =  925   composites

Angle  B  in the smaller( isosceles) triangle =   (180 -70) / 2  =  55°

So we can find the  base of the larger triangle as

base  /sin70  = 27/sin55

base =  27*sin 70 / sin 55  ≈  31

So

x =  31/sin27  ≈  68.3°

a)   the tangent is negative in this quadrant     (true)

b)   the sine is positive in this quadrant  so the cosecant   is also positive  (false)

I'm assuming that the height of th balcony is  4m

So....the distance between them  is  given by

4 /tan (52°)   +  4/tan 47°)   ≈   6.86 ft

hello.

The sine rule says that

\(\displaystyle \frac{a}{\sin A} = \frac{b}{\sin B}=\frac{c}{\sin C} = 2R,\)

(where R is the radius of the circumscribing circle), so

\(\displaystyle a=2R\sin A,\\b = 2R \sin B,\\ c = 2R\sin C,\)

and since 

\(\displaystyle \sin A :\sin B :\sin C \)

are in the ratio 5 : 6 : 7, it follows that a, b, and c are also in the ratio 5 to 6 to 7.

Since a + b + c = 18 it then follows that a = 5, b = 6 and c = 7.

From that, s (the semi-perimeter) = 9, so the area of the triangle is

\(\displaystyle \sqrt{9(9-5)(9-6)(9-7)}=6\sqrt{6}.\)

n.b. the angles, (in degrees to 3dp) are 44.415, 57.122 and 78.463.

5000 mod (13 - 1 ) ==8

OR

5000 mod(25 - 1) == 8

The computer says the 5000th integer on list is 8:

1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
10 10
11 11
12 12
13 13
14 12
15 11
16 10
17 9
18 8
19 7
20 6
21 5
22 4
23 3
24 2
25 1
26 2
27 3
28 4
29 5
30 6
31 7
32 8
33 9
34 10
35 11
36 12
37 13
38 12
39 11
40 10
41 9
42 8
43 7
44 6
45 5
46 4
47 3
48 2
49 1
50 2
51 3
52 4
53 5
54 6
55 7
56 8
57 9
58 10
59 11
60 12
61 13
62 12
63 11
64 10
65 9
66 8
67 7
68 6
69 5
70 4
71 3
72 2
73 1
74 2
75 3
76 4
77 5
78 6
79 7
80 8
81 9
82 10
83 11
84 12
85 13
86 12
87 11
88 10
89 9
90 8
91 7
92 6
93 5
94 4
95 3
96 2
97 1
98 2
99 3
100 4
101 5
102 6
103 7
104 8
105 9
106 10
107 11
108 12
109 13
110 12
111 11
112 10
113 9
114 8
115 7
116 6
117 5
118 4
119 3
120 2
121 1

.

.

.

4993 1
4994 2
4995 3
4996 4
4997 5
4998 6
4999 7
5000 8

x(x+5) = -n

\(x(x+5) = -n\\ x^2-5x+n=0\\ x=\frac{5\pm\sqrt{25-4n}}{2}\\ \)

this will NO real solutions if the number under the square root is negative.

\(25-4n<0\\ 25<4n\\ 4n>25\\ n>6.25\)

So there will be no real solution if n = 7,8,9 or 10

that is  4/10 = 2/5 probability

There is no second one.

ok I answered it.

There were some avocados in 3 boxes, E, F and G. 30% of the number of avocados in Box E was equal to 60% of the number of avocados in Box F. The number of avocados in Box G was 60% of the total number of avocados in Box E and F. After Oscar removed 20% of the avocados in Box G, there were 88 more avocados in Box F than in Box G. In the end, how many avocados should be transferred from Box F to Box G so that the avocados in Box G would be the same as Box E?

I do not think that this scenario is possible.

30% of the number of avocados in Box E was equal to 60% of the number of avocados in Box F

so

0.3E=0.6F

E=2F

So in the beginning there are 2 times more in box E than in box F     (And  each one has a muliple of 10)

G=0.6(E+F)

G=0.6(3F)

G=1.8F

Now 20% are removed from box G

so box G will then have    0.8*1.8F = 1.44F

So now

It is clear that there are now mow avocados in the last box than in the middle one.

So there cannot be 88 more in the middle one, that is a contradiction.

A  =   (5/18)*180  =   50°

B =    (6/18) * 180 = 60°

C  =   (7/18)* 180 =  70°

By the Law of Sines

a /sin 50  = b / sin 60                 c / sin 70  = b / sin 60

a / b =  sin 50 /sqrt(3)/2             c/b = sin 70 / sqrt (3)/2 

a =  b  [ 2sin 50  / sqrt 3]            c =   b [ 2sin 70 / sqrt 3 ]

a  +  b +  c  =   180

b* 2sin 50 /sqrt 3    + b  +  b [ 2sin 70 /sqrt 3 ]   =  18

b  [  2sin 50 /sqrt 3  + 1  + 2sin70/sqrt 3 ] =  18

b =  18  /  [ 2sin50/sqrt 3  + 1 + 2sin 70 / sqrt 3 ]  ≈  6.06

a = 6.06 [ 2 sin 50 / sqrt 3]  ≈ 5.36

c =  6.06 [2 sin 70 /sqrt 3 ]  ≈ 6.58

Area =  sqrt  [ 9  (9 - 6.06) (9 - 5.36) (9 - 6.58)  ]  ≈   15.267  units

1/3 =  10/30  =  

11/12 =   27.5 / 30 

So

[ 27  -11 ] + 1  =    17 integer values

Yep got it. Thanks everyone!

Of course π/10+3π/10 =2π/5, not π/5, sorry! So a+b=2+5=7, as CPhill said.