Solve for real x the equation x=6[(x-2)^1/2-1].

For the beggining, we'll impose constraint of existence of
the square root:

x - 2 >=
0

x >= 2

The real
solutions of the equation have to belong to the range [2 ;
+infinite).

Now, we'll solve the equation. We'll start by
removing the brackets:

x = 6sqrt(x-2) -
6

We'll add 6 both sides, in order to isolate the square
root to the right side.

x + 6 =
6sqrt(x-2)

Now, we'll square raise to eliminate the square
root;

x^2 + 12x + 36 = 36x -
72

We'll subtract 36x -
72:

x^2 + 12x + 36 - 36x + 72 =
0

We'll combine like
terms:

x^2 - 24x + 108 =
0

We'll apply quadratic
formula:

x1 =[24 + sqrt(576 -
432)]/2

x1 = (24 + 12)/2

x1 =
18

x2 = (24 - 12)/2

x2 =
6

Since both values belong to the interval of
admissible values, we'll accept them. The solutions of the equation are: {6 ;
18}.

Write as partial fractions : [5x^2 + 10 x + 3] / [ x^3 + 2x^2 – x – 2]

We have to express [5x^2 + 10 x + 3] / [x^3 + 2x^2 – x –
2] as partial fractions.

First, we factorize the
denominator

[5x^2 + 10 x + 3] / [x^3 + 2x^2 – x –
2]

=> [5x^2 + 10 x + 3] / [x^2(x + 2)-1(x +
2]

=> [5x^2 + 10 x + 3] / [(x^2 – 1) (x +
2)]

=> [5x^2 + 10 x + 3] / [(x – 1) (x + 1) (x +
2)]

Let us write the given expression as A / (x – 1) +B /(x
+ 1) + C/(x + 2)

=> [A(x +1) (x +2) + B(x – 1) (x
+2) + C(x – 1) (x +1)]/ [(x – 1) (x + 1) (x + 2)]

=>
[A(x^2 + 3x + 2) + B(x^2 + x – 2) + C(x^2 – 1)] / [(x – 1) (x + 1) (x +
2)]

[A(x^2 + 3x + 2) + B(x^2 + x – 2) + C(x^2 – 1)] / [(x –
1) (x + 1) (x + 2)] = [5x^2 + 10 x + 3] / [(x – 1) (x + 1) (x +
2)]

=> [A(x^2 + 3x + 2) + B(x^2 + x – 2) + C(x^2 –
1)] = [5x^2 + 10 x + 3]

=> Ax^2 + 3Ax + 2A + Bx^2 +
Bx – 2B + Cx^2 – C = 5x^2 + 10 x + 3

Equating the
coefficients of x^2, x and the numeric term we get:

(A + B
+ C) = 5 … (1)

(3A + B) = 10 …
(2)

(2A – 2B – C) = 3… (3)

Add
(1) and (3)

=> 3A – B =
8

Now 3A – B = 8 and 3A + B =
10

=> 6A = 18

=>
A = 3

B = 3A – 8 = 9 – 8 = 1

C
= 5 – A – B = 5 – 3 – 1 = 1

Therefore we get A / (x – 1) +B
/(x + 1) + C/(x + 2) = 3 / (x – 1) +1 /(x + 1) + 1/(x +
2)

The given expression written as partial fractions
is:

3/(x – 1) +1/(x + 1) + 1/(x +
2)

In "The Rime of the Ancient Mariner," how was the albatross received?

In the middle of a terrible and frightening situation,
when the ship has been blown off-course and is surrounded by fog and massive, creaking
icebergs that are mast-high, the albatross is the first living thing that the sailors
see in their new environment near the South Pole. Birds, of course, usually mean that
land is somewhere near, and so are greeted particularly warmly, but especially for the
sailors who have been at sea during an ice storm, the albatross is greeted with open
arms.

Key to note is the distinctly religious way that it
is introduced. The albatross "crosses" the ship and is described as if it were a
"Christian soul." The albatross seems to bring the sailors good luck, as after its
arrival the helmsman "steers us through" and a good south wind gives the ship the motion
it needs to continue on its journey. The albatross thus becomes a firm favourite of the
sailors:

And
a good south wind sprung up behind;

The Albatross did
follow,

And every day, for food or
play,

Come to the mariner's
hello!

In mist or cloud, on mast or
shroud,

It perched for vespers
nine;

Whiles all the night, through fog-smoke
white,

Glimmered the white
Moonshine.

Note how the
Albatross becomes a firm fixture of life on the ship. He always joins the crew for their
evening prayers and is compared to "white Moonshine," clearly referring to the luck that
the sailors believe it brings them.

In "To Build a Fire," what are the three things the dog knew by instinct?

The dog's instincts tell him that it is too cold to be out
on the trail, a truth that eludes the man until it is too late. The dog also knows that
he must avoid getting wet, and when he does get his paws wet, he instinctively bites the
ice away to keep them from freezing. When the man plans to kill the dog, the animal
senses the fear in his voice and behavior and eludes him. Finally, the dog knows
instinctively that fire and its warmth mean security and survival. When the man fails to
rekindle his fire and consequently freezes to death, the dog leaves him to go find
another fire. The dog's instincts in this natural environment proved to be far superior
to the man's human intelligence and faulty judgment.

1) a)describe "Banquo's revenge" in act 4 scene 1b) what is its effect of Macbeth

Concerning Shakepeare's Macbeth, I
assume you're referring to the vision Macbeth sees that presents Banquo's heirs as
kings. The vision presents the line of Stuart kings which culminates in James I, who was
reigning at the time the play was first produced. 

Macbeth,
of course, tries to change fate in Act 3 when he plans and orders the killing of Banquo
and Fleance.  He succeeds in eliminating Banquo, but fails to eliminate Fleance.  This
makes the ruling by Banquo's heirs possible.

The last
figure shown in the vision is Banquo, holding a magic mirror that shows Macbeth many
more kings.  Macbeth yells, "I'll see no more."  But the kings just keep coming.  And
the blood-matted Banquo smiles at Macbeth while pointing to the
kings. 

Macbeth asks the witches if this is
so. 

These are the only specific reactions to the vision
revealed.  But, in reaction to all of the visions shown him, and in reaction to the news
that Macduff has fled to England, Macbeth determines to act immediately on his first
thoughts:

The
very firstlings of my heart shall be

The firstlings of my
hand.

He will not think or
hesitate before acting from now on.  And he keeps his word, immediately ordering the
slaughter of Macduff's family.  

How does the answer he gives to the Sphinx's riddle foreshadow Oedipus's own life?Oedipus Rex by Sophocles

The irony of Oedipus's life in Sophocles Oedipus
Rex is that he intends to find the truth and instead meets his fate.  When a
drunken man tells Oedipus he is not his father's son, Oedipus seeks the truth by
traveling to consult a Dephic oracle; however, the oracle says merely that Oedipus would
kill his father and beget children by his mother. Resolved to avert such a fate, Oedipus
determines to never return to Corinth where he has grow up with Polybus and Morope, the
king and queen.  Unfortunately for Oedipus, along the way from Delphi, he becomes
angered at a man who orders him off the road, killing him.  Later, outside Thebes,
Oedipus encounters the Sphinx and solves its riddle.  As a reward, he is made King of
Thebes and is given the hand of Jocasta in marriage.  Thus, he has fulfilled the
prophecy of the oracle, ironically, because he has made such efforts to prove him
wrong. 

Indeed, it is this pride, or hubris,
which seals the fate of Oedipus the king. His burning demand to know the
truth brings him to Delphi; his return from Delphi effects his encountering of the man
in the chariot who--unbeknowst to Oedipus--is his father, and it is his insistent pride
that causes Oedipus to demand the truth from Teiresias and Creon so that he can rid
Thebes of its plague.  Finally, with more irony, Oedipus seals his own fate as he calls
for an investigation of Laius's death, an investigation that will not "bring what is
dark to light," but, rather, bring about Oedipus's blinding, as in guilt and agony,
Oedipus removes his own eyes.

As in Robert Frost's poem,
"The Road Not Taken," Oedipus takes the path that seems the right one,
but

knowing
how way leads on to way,

he
cannot turn back and not know what he knows now.  His life is altered beyond anything he
ever has imagined.

Do you think Hamlet and Ophelia are a good match in Hamlet?

For a textual answer to your question you could look at
what both Laertes and Polonius say to Ophelia in Act 1 of the play.  Laertes and Ophelia
have an interesting conversation right before Laertes plans to go back to school.  As
the attentive and loving big brother, Laertes asks Ophelia about her relationship with
Hamlet, and tells her that he thinks that, while Hamlet may express that he loves her,
she shouldn't take him too seriously.  He warns her
to

Hold it a
fashion, and a toy in blood; . . .

Forward, not permanent
-- sweet, not lasting.

When
Ophelia questions that assumption, Laertes gives her a very solid assessment of the
situation.  He explains that, as a prince, Hamlet is not as free as other men are to
choose their mates.  He's right when he says

readability="21">

Perhaps he loves you
now,

And now no soil nor cautel doth
besmirch

 The virtue of his will; but you must
fear,

His greatness weighed, his will is not his
own,

For he himself is subject to his birth. . .
.

for on his choice [of wife
]depends

The safety and heath of his whole state
[Denmark]

Generally, princes
marry princesses for social or political gain.  Ophelia is not a princess.  Hamlet may
have true love for Ophelia, but that doesn't guarentee that he can choose her as his
wife.  He cautions her to not lose her honor to Hamlet and lose her heart and her
reputation.  Given these ideas, it would not seem that Hamlet and Ophelia are not a good
match. 

On the other hand, Ophelia expresses genuine
affection for Hamlet and defends Hamlet and their relationship when her father has a
similiar conversation with her about Hamlet.  Polonius isn't as diplomatic about
Hamlet's motives, suggesting something a bit more base, but his point is still the
same.  Hamlet is a prince, and as such, lives a very different life, with very different
expectations. 

There is textual evidence that Ophelia
thinks highly of Hamlet and that she is legitimately concerned over the apparent madness
of Hamlet as it is displayed later in the play.  Consider her prayers in Act 3
when Hamlet is berating her in the "get thee to a nunnery scene.  She is so devastated
by his killing her father and his rejection of her that she loses her sanity and ends up
killing herself.  Her feelings for Hamlet are true, and as evidenced by Hamlet's
reaction to her death in Act 5 his are as well, but the circumstances of their lives in
the time frame of this play strain the relationship beyond its
endurance.