How is the building of the church by John Merrick a central metaphor in the play the Elephant Man?

Merrick struggles throughout the play to find a normal
life, given the enormous growths on his head that lead to him being part of a freak show
for much of his early life.  He is taken advantage of frequently and struggles to have
real friendships and a sense of purpose.

But he has very
deep religious beliefs, and part of the way he expresses them is in the way he studies
and eventually constructs a model of St. Phillip's church.  The model building coincides
with his progression and the fact that he has finally found some human companionship and
has found people who are willing to listen to him and consider him
human.

As he continues the construction of the church, his
understanding of and relationships with the people around him deepen and he appears to
be moving towards a somewhat complete life including a sexual awakening through his
relationship and conversations with Mrs. Kendal.

He
finishes the model just before he dies, and in some ways the model might then be
interpreted as a model of his life.  As it reaches a completed state, his purpose and
the conflict are resolved and he is released from the difficulties of his mortal body
and its disfigurements.

In Macbeth, why did Lady Macbeth have anxiety attacks and could not sleep?

Sorry to begin by correcting your question, but Lady
Macbeth's problem was not that she could not sleep. Rather she could, but when she did
sleep, she would sleepwalk and talk about what she had done. It becomes evident if you
have a look at what she says in Act V scene i that her problem is caused by massive
guilt about what she has personally done and now what Macbeth, her husband, has gone on
to do now that she has pointed him in the right direction of rapacious ambition that
will let nothing stand in its way. Consider what she says in this scene to her unknown
audience of the Doctor and Gentlewoman:

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Out, damned spot! out, I say!--One; two: why,
then 'tis time to do't.--Hell is murky.--Fie, my Lord, fie! a soldier and afeard?--What
need we fear who knows it, when none can call our power to accompt?-Yet who would have
thought the old man to have had so much blood in
him?

Notice how her speech
starts with trying to rid her hands of the stain of blood after she has taken the dagger
and covered the poor henchman with Duncan's blood. It is clear that she is re-living a
moment that now is particularly traumatic to her, and the guilt that she is experiencing
is particularly evident in the last line, as, in spite of all of her tough talk, the
amount of blood that emerges after the deed has been done shocks
her.

Prove that the iidentity is true in any triangle: b*cosC+c*cosB=a

We'll prove this with the help from the cosine
theorem:

b^2=c^2+a^2-2ac*cosB,  cos B =
(c^2+a^2-b^2)/2ac

c^2=a^2+b^2-2ab*cosC,  cos C =
(a^2+b^2-c^2)/2ab

Now, we'll substitute cos B and cos C, by
the found expressions, so
that:

b*cosC+c*cosB=a

b*(a^2+b^2-c^2)/2ab
+ c*(c^2+a^2-b^2)/2ac = a

After reducing the terms, we'll
obtain the irreducible quotients:

(a^2+b^2-c^2)/2a +
(c^2+a^2-b^2)/2a = a

We'll multiply the term from the right
side, by 2a:

a^2+b^2-c^2+c^2+a^2-b^2 =
2a^2

After reducing similar terms, we'll
obtain:

a^2+a^2 = 2a^2

2a^2 =
2a^2

The last result tells us that the identity,
b*cosC+c*cosB=a, is valid for any triangle.

Please provide an example of how a creepy mood is created in "The Tell-Tale Heart."EXAMPLE: Description of the old man's eye--the eye of a vulture....

It is clear that in this masterful short story, Poe
creates an atmosphere and mood of terror and fear. Certainly there are plenty of ways in
which he does this, but you might want to consider a description from the middle of the
story, whilst the narrator is in the room of the old man and is contemplating his eye,
waiting to kill him. Suddenly, he is able to hear the heartbeat of the old man, thanks
to the acuteness of his senses he has:

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Meaning the hellish tattoo of the heart
increased. It grew quicker and quicker, and louder and louder every instant. The old
man's terror must have been extreme! It grew louder, I say, louder every
moment!

Consider how the
description of the "hellish tattoo" of the old man's heart that the narrator is able to
hear, and how it becomes louder and quicker every second, helps raise suspense for what
is going to happen and also conveys the severe terror of the old man, who knows that
someone is in the room with him and is terrified. Thus the mood of fear and terror is
sustained in this section of the short story.

In I Am Legend, how and why is the protagonist a legend?

Legends have been defined as historically grounded fairy
tales, folk tales and historical narratives with embellishments. A person is a legend if
they stand out or if they perform incredible tasks. Robert Neville is a legend because
he does stand out. In the film version, he becomes legendary after his death for his
contribution to the cure. But I think in the context of the book, he is a legend because
he represents the last of the human race prior to the spread of the vampirism
virus.

Legends are often based upon actual events. Over
time, they are embellished. Sometimes, they are embellished to the point that they
become verisimilar (only resembling truth). If they are further embellished, they become
mythological. In this case, the tale may be based on historical events, but it has
become so impossible to believe that it becomes a myth.

I
don’t think Neville’s case has become myth. It is still verisimilar even as a fiction.
It is possible that a virus could spread resulting in this apocalyptic scenario. It is
unlikely but possible. So, I would say that Neville is a legend because he does stand
out in his own time. Until he meets Ruth, he is alone in a world of zombies. He becomes
“legend” when he dies because he is the last of his kind. He is the last of the old
world of humanity. That old world has now become legend.

How are others becoming suspicious of Macbeth's murder of Duncan?

In Shakespeare's Macbeth, two main
events lead to others in the play suspecting Macbeth of
treachery.

First, Macbeth kills the two grooms that could
have been witnesses of Duncan's murder.  In Act 2.3 Lennox states that the grooms seem
to have done the murder, since their daggers and hands and faces were covered in blood. 
The thanes present would of course want to interrogate them, but Macbeth makes that
impossible, announcing:

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Oh, yet I do repent me of my
fury,

That I did kill them.  (Act
2.3.103-104)

Though no one
else says anything at the time, Macduff realizes the problem with Macbeth's killing of
the grooms:

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Wherefore did you so?  (Act
2.3.104)

Macbeth tries to
explain his actions and play it off, so to speak, but Macduff is not convinced.  When
the time comes for Macbeth's coronation as king, instead of going to witness and
celebrate with Macbeth, Macduff instead goes home "to Fife (Act 2.4.37), further
revealing his suspicions.  And, of course, Macduff turns out to be directly involved in
Macbeth's downfall.

Second, the killing of Banquo and the
attempted killing of Fleance then lead to widespread suspicions.  In Act 3.6 Lennox and
an unidentified Lord speak sarcastically and ironically about the coincidences of
Duncan's and Banquo's deaths, as well as the deaths of the two grooms.  By this point,
too many murders have occurred for others to not suspect Macbeth of
treachery. 

Adding to the murders, is the fact that,
apparently, Macbeth has become a tyrant.  Lennox refer's to the "tyrant's feast," and
the Lord refers to Macbeth as "this tyrant."  The two hope Macbeth will be overthrown,
so that

...we
may again

Give to our tables meat, sleep to our
nights,

Free from our feasts and banquets bloody
knives,...  (Act
3.6.33-35)

Thus, Macbeth's
tyranny has made others in the play discontented, and the string of murders has made
them suspicious.

Ironically, even Lady Macbeth almost
immediately adds to the suspicion after Macbeth kills the grooms.  Macbeth veers from
her plan when he kills the grooms and Banquo, as stated.  He'd have been much better off
following her directions.  But even she reveals evidence of their guilt, even though
Banquo seems to be the only one who catches it.  When she is told of Duncan's death, she
responds:

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What, in our house?  (Act
2.3.84)

And Banquo
understands the inappropriateness of her response and rebukes
her:

Too cruel
anywhere.  (Act
2.3.84)

Banquo understands
that there is something warped about a woman whose first thought after hearing of a
king's assassination is that it reflects badly on her hospitality.  Banquo never acts on
his suspicions, but they are certainly present in his mind. 

the quadratic equation 3x^2-7x=3 has roots that are...1.)real, rational, and equal 2.)real, rational and unequal 3.)real, irrational and unequal...

Given the quadratic
equation:

3x^2 - 7x =
3

==> 3x^2 - 7x -3 =
0

We need to determine the type of
roots.

Then, we will use the discriminant to find
out.

We know that:

delta = b^2
- 4ac

If delta = 0 ==> then the equation has one
real root.

If delta > 0 ==> then the equation
has two real roots.

If delta < 0 then the equation
has two complex roots.

If delta is a complete square, then
it has 2 rational roots

If delta is not a complete square,
then it has 2 irrational roots.

Let us test
delta.

delta = b^2 -4ac = (-7)^2 - 4*3*-3 = 49 + 36 =
85

Then delta > 0 , then it has two real
roots.

Also, delta is NOT a complete square, then it has
irrational roots.

Then, the answer is number
(3) Real, irrational, and unequal.