College Algebra, Chapter 1, 1.5, Section 1.5, Problem 2

Solve the equation $\sqrt{2x} + x = 0$ by doing the following steps.

a.) Isolating the radical.


$
\begin{equation}
\begin{aligned}

\sqrt{2x} + x =& 0
&& \text{Given}
\\
\\
\sqrt{2x} =& -x
&& \text{Isolate the radical}
\\
\\
\sqrt{2} \cdot \sqrt{x} =& -x
&& \text{Divide both sides by } \sqrt{x}
\\
\\
\sqrt{2} =& \frac{-x }{\sqrt{x}}
&& \text{Apply the properties of exponent}
\\
\\
\sqrt{2} =& -x^{\left( 1 - \frac{1}{2} \right)}
&&
\\
\\
\sqrt{2} =& -x^{\frac{1}{2}}
&& \text{Square both sides}
\\
\\
(\sqrt{2})^2 =& (-x^{\frac{1}{2}})^2
&& \text{Simplify}
\\
\\
x =& 2

\end{aligned}
\end{equation}
$


b.) Squaring both sides


$
\begin{equation}
\begin{aligned}

\sqrt{2x} + x =& 0
&& \text{Given}
\\
\\
(x) =& (- \sqrt{2x})^2
&& \text{Subtract } \sqrt{2x}
\\
\\
x^2 =& 2x
&& \text{Square both sides}
\\
\\
x^2 - 2x =& 0
&& \text{Subtract } 2x
\\
\\
x( x - 2) =& 0
&& \text{Factor out } x
\\
\\
x =& 0 \text{ and } x - 2 = 0
&& \text{Zero Product Property}
\\
\\
x =& 0 \text{ and } x = 2
&& \text{Solve for } x

\end{aligned}
\end{equation}
$


c.) The solutions of the resulting quadratic equation are ______.

The solutions of the resulting quadratic equation are $x = 0$ and $x = 2$.

d.) The solution(s) that satisfy the original equation are ______.

The solution(s) that satisfy the original equation are $x = 0$ and $x = 2$.

How did the United States dehumanize Germany in World War I?

The United States was able to dehumanize Germany during World War I. Germany thought it would win this war, and when Germany lost many people couldn’t believe that they had been defeated. The Germans had spent much time, effort, and money to build up its military in order to gain colonies. Germany thought its military was the best in the world and would win any conflict in which it fought. Germany was convinced it would defeat the Allies before the United States would become a factor in the war. This is one reason why the Germans began to sink ships again without warning them, even though Germany knew this would bring the United States into the war on the side of the Allies.
During the war, the people of Germany suffered greatly. The German economy was based on trade, and a prolonged war impacted that trade significantly. As the war dragged on, there were shortages of basic products that people needed such as meat, bread, dairy products, and potatoes. This led to riots within Germany in 1916. The government took more control over the economy, and the army influenced economic policy.
When Germany was defeated, very harsh peace terms were forced upon Germany. Germany lost some land and was unable to have a military that could go on the offensive. Germany also had to pay heavy reparations to the Allies. These harsh terms were imposed on Germany even though the United States suggested to Germany that the peace treaty would be an easier one on Germany if Germany surrendered instead of fighting to the bitter end. The Germans felt that they were duped by this suggestion, which helped lead Adolf Hitler to later call for revenge.
After the war, conditions became very desperate in Germany. There was widespread unemployment, and inflation soared. Many Germans suffered greatly. This helped propel a man with extreme views to eventually take control in Germany. This man, Adolf Hitler, led Germany on a path the created much death, destruction, and devastation as a result of the eventual outbreak of World War II in 1939.
https://alphahistory.com/weimarrepublic/world-war-i/

https://www.firstworldwar.com/features/germanyduringww1.htm

What is the relationship between the trial of Tom Robinson and the Boo Radley subplot in the novel?

In To Kill a Mockingbird, both Tom Robinson and Boo Radley are misunderstood characters that are not guilty of the "crimes" they are accused of committing. In the case of Tom, he feels sorry for Mayella Ewell and stops to assist her with her work. She and her father accuse him of raping her. Her father has to create a way to explain the bruises he gives to her, and when he finds her with a black man, he sees the perfect opportunity. Tom can be considered a "mockingbird." Early in the novel in Chapter 10, Atticus says, "it's a sin to kill a mockingbird." Tom is found guilty because Maycomb County jurors side with a white man over a black man.
Boo Radley, also a "mockingbird," is another character that is unjustly punished primarily for being misunderstood. Boo is confined to stay inside his house first by his father, then by his brother. He is the topic of many neighborhood superstitions and tales. For example, Miss Stephanie Crawford believes he was outside her window once looking in on her. The children are scared of being too close to the Radley property but by the end of the story, they begin to recognize that Boo's intentions are kind. In Chapter 3, Atticus explains to Scout that "You never really understand a person until you consider things from his point of view." Scout matures throughout the novel and begins to consider her father's words, especially in the case of Boo.
Near the end of the novel, the two stories merge when Bob Ewell attempts to kill Scout and Jem. Bob is angry because Atticus makes it clear that he is the one that injures his own daughter. Boo Radley intervenes and kills Bob Ewell in the process. The sheriff and Atticus agree that it would be wrong to drag Boo into the spotlight, so they decide to stick with the story that Bob fell on his own knife. Scout agrees and says, "it'd be sort of like shootin' a mockingbird."

Travis kills Eckels at the end of the story. Why does he do this?

Travis kills Eckels at the end of the story because Eckels disobeyed the rules of traveling back into time to the dinosaur age—but as we will see, Bradbury implies it is more complicated than that.
Eckels panicked on the safari to the dinosaur age and left the special path constructed to safeguard against any interference with the past that could alter the course of history. In the process, he stepped on and smashed a butterfly: a seemingly minor act, but one that changed the future in profound ways.
We know the future is different when the safari party returns because common English spellings have changed, and more importantly, now Deutscher, not Keith, has won the recent presidential election. As a worker says:

We have got an iron man now, a man who isn't afraid of anyone or anything!"

The piece on information that a would-be dictator has been elected, along with knowing he is responsible for it, anguishes Eckels. Eckels lets out a cry and falls on his knees. This complicates Travis's possible motivations for killing Eckels: the sequence of events suggests that Travis kills Eckels as much out of mercy—pity for his guilt and agony—as for punishment.

Ah, but does Travis kill Eckels? We only know that the final “sound of thunder” is probably a gunshot. Travis could kill Eckels. Or he could kill himself. Obviously, Eckels accidentally changes the future when he steps off the path and onto a butterfly. He has been told the rules and the need for them to be followed. But the dinosaur so terrifies him, that he is lucky enough to even make it back to safety of the time machine. The group cannot go back and repair the damage he has done. And now people have to live with a dictator as a president, instead of a compassionate leader. Someone has to pay the price for the mistake. If Travis kills Eckels, he’s issuing the ultimate penalty for the man’s blunder. If Travis kills himself, he won’t have to live under this new regime, and he won’t have to answer to authorities about the fault of Time Safari to use time travel wisely. But Eckels, in turn, would have to live with his mistake. Which fate would be worse?

Did Mein Kampf have anything to do with the techniques Hitler would use to rise to power?

Hitler mainly used Mein Kampf to outline the ideological foundations of what became known as the National Socialist Workers' Party, or Nazis. In this sense Mein Kampf was itself a tool for Hitler's rise, as it was quite widely read in the early 1930s, as Nazis were beginning to win seats in the Reichstag and Hitler was becoming a nationally prominent politician. In Mein Kampf Hitler elucidates his views on the so-called "stab in the back" that led to German surrender in World War One, his virulent brand of anti-Semitism, and his bizarre racial theory of history. He did describe, in great detail, how propaganda should be used to facilitate the Nazi rise to power. Propaganda, he told his readers, was not intended to "weigh and ponder the rights of different people." Rather, its purpose was to:

...emphasize the one right which it has set out to argue for. Its task is not to make an objective study of the truth, in so far as it favors the enemy, and then set it before the masses with academic fairness; its task is to serve our own right, always and unflinchingly.

So Mein Kampf emphasized the importance of propaganda, the centrality of race, and the supposed need for totalitarian leadership to restore Germany. In this way, it anticipated the methods that Hitler would use in his rise to power and many aspects of his policies once he seized control of Germany.
https://www.jewishvirtuallibrary.org/excerpts-from-mein-kampf

Why is the Reconstruction era considered a revolution, and why does Foner say it is unfinished?

The Reconstruction Era began a revolution in the freedom of former slaves. They experienced new elements of freedom, such as the rights of citizenship given in the 14th Amendment. African-American men were given the right to vote with the 15th Amendment. In addition, at least in theory, they were freer to control their own movements, marry, and join the church and community organizations of their choosing.
On the other hand, the reality of Reconstruction was that the promise of complete freedom for former slaves was not fulfilled and the revolution, as Foner writes, was unfinished. For example, while some freed slaves were promised land, not many were able to attain land ownership and instead worked as sharecroppers on others' land. In addition, the right of African-Americans to vote was often curtailed in the later years of Reconstruction and afterward. The revolution that began during Reconstruction continued into the 20th century and beyond. 

How did the enemies of the Aztec help the Spanish to defeat the Aztec?

The Aztec Empire, which flourished between 1420 and 1520 approximately, dominated many of the neighboring populations. Some were allies, but many remained resentful enemies, even though they were subjugated. Some, like the Tlaxcalan, remained independent, despite the fact that the Aztec tried to conquer them many times.
When Cortes arrived in Mexico in 1519, he had to pass through the lands of many of the Aztec enemies, such as the Tlaxcalan. Cortes managed to make allies of the Tlaxcalan and of other populations by promising them freedom from the Aztec and giving them gifts. He understood that he needed the support and military strength of other populations who knew the land and the Aztec better. He also needed a safe base in which to rest and recover after battles.
With the help of these neighboring tribes, the Spanish easily defeated the Aztec. Cortes quickly understood that the internal resentment which many tribes harbored against the Aztec could be useful to the Spanish, and he used it to their advantage.