TheXSquaredFactor

If it is difficult for you to comprehend this problem, then it may be best to think of an example. Don't overcomplicate matters, though! I think \(\frac{x^3+5x^2}{x}=0\) is an adequate example because this equation involved the elimination of a variable. 

If you were to solve it, it may look something like this!

\(\frac{x^3+5x^2}{x}=0\\ \frac{x^2(x+5)}{x}=0\\ x(x+5)=0\\ x=0\text{ and }x=-5\)

A) Confirming the validity of all roots is essential. Notice that \(x=0\) in the previous example is an extraneous solution because it results in a division-by-zero error.

B) Factoring was essential to recognize that there was a common factor in the first place!

C) Yes, this is a necessary precaution. If we set the common factor equal to zero, then we will get that \(x\neq0\) because this does not fit the domain.

D) This step is illogical; do not do this.

Hi, ManuelBautista2019!
 

The discriminant of a quadratic equation is the expression of the radicand of the quadratic formula \(x ={-b \pm \sqrt{\textcolor{red}{b^2-4ac}} \over 2a}\). In the given quadratic equation x^2-5x+10, a=1, b=-5, and c=+10. The discriminant will allow you to answer every question here.

Let's see the value of the discriminant and interpret its meaning. 

\(x^2-5x+10\\ a=1,b=-5,c=10\\ b^2-4ac\\ (-5)^2-4*1*10\\ 25-40\\ -15 \) 

1) The discriminant of the original quadratic yields a negative value, as calculated above, so the roots are imaginary.

2) The roots of a quadratic can be equal if and only if the discriminant equals 0, so the roots are unequal.

3) We already determined that the roots are imaginary (look at #1), so the roots are neither of the options listed. 

The question asks about the type of roots--not what the roots are. 

Because of this technicality, we can combine knowledge from Descartes' Rule of Signs and the Fundamental Theorem of Algebra and the Complex Conjugate Root Theorem in order to answer this question. 

The Descartes' Rule of Signs states that the number of sign changes in \(f(x)\), when the terms are ordered by degree, indicates the number of positive solutions or less than this by an even number. Let's consider the original function \(f(x)=5x^2-7x-6\):

\(f(x)=\underbrace{5x^2-7}\underbrace{x-6}\\ \hspace{16mm}+1\hspace{7mm}+0\hspace{10mm}=\text{1 sign change}\)

Notice that +5x² and -7x have a different sign, so I marked that change with a +1. -7x and -6 do not have a signage change, so I marked it with a +0.

The Fundamental Theorem of Algebra is relevant because it indicates how many total solutions that a function can have. In this case, there are 2 roots since the degree of the polynomial is 2. 

The Complex Conjugate Root Theorem states that if polynomials have real coefficients with complex roots, then they will come in pairs; in other words, there will always be an even number of non-real roots.  

I would say that it would be wise to consolidate this information into a table.

Based on the rules mentioned previously, only one possibility remains: Both roots are real.

Those are some pretty diagrams, CoopTheDupe!

I am fairly certain that you were referring to the Triangle Proportionality Theorem when you stated "MN || PR -> Same Side Rule (Forgot what it's called exactly)"

This puzzle has a very elegant solution to it!
 

Actually, I think the best way to think about this one is to think two-dimensionally first! We have 3 points that we place on a circle! 

I have picked arbitrary points for A, B, and C. I want to determine the probability when the three points form a triangle wherein the center is included. No matter what three points you pick, there are four options

Option 1: Point A, Point B and Point C are where they are in the diagram.

Option 2: Point A is moved to the other side of line, Point B stays and Point C stays

Option 3: Point A stays, Point B moves to the other side of line, and Point C stays

Option 4: Point A moves to the other side of the line, Point B moves to the other side of the line, and Point C stays

In these 4 cases, only one of the situations has a polygon that includes the center, so for a circle, the probability is \(\frac{1}{4}\).

Actually, let's go down to one dimension! Let's just worry about a segment.

There are two possibilities.

1) Point B is on the left of O

2) Point B is on the right of O

In this case, there is 1 case that results in the center being included, so the probability is \(\frac{1}{2}\).

If a one-dimensional case yielded \(\frac{1}{2^1}\) and the two-dimensional case yielded \(\frac{1}{2^2}\), then it would seem logical to me to assume that the three-dimensional case would be \(\frac{1}{2^3}=\frac{1}{8}\). In fact, if you try the cases again, you will see that this is the case.

602.88cm³ is correct!

You're learning well!

Mount Everest?

Here is my rationale!

1) This infamous mountain is buried underneath that Earth crust!

2) The mountain towers into the sky, and reaches an incredible altitude of 29,000 feet!

3) I believe the most recent statistic states that the mountain has claimed over 200 experienced yet valiant climbers! The air at that altitude of 29000 feet is gruesomely and lethally thin--even with the aide of supplemental oxygen tanks. 

I think this question is one where some people might want to pull out their hair in frustration. I suggest gathering all your information into an organized table. 

We may not be able to fill in all the information, but that is OK! Do not forget what the end goal is! The end goal is to figure out how much time it will take if Paulo and Irina work together. This is the information we know:

• Irina painted 1 room
• Irina painted that room in 9 hours
• Paulo painted the same room
• Paulo painted that room in 8 hours
• Together, Paulo and Irina will paint one room

Now that we know all this information, let's fill in the table. 

Now, what is the painting speed of Paulo? I think you would agree that Paulo can paint at 1 room per 8 hours. This is the rate for Paulo! We can use the same logic for Irina. Irina paints at a speed of 1 room per 9 hours. Let's fill that information in!

In order to fill in the rest of the table, we will have to introduce variables! I will use "x" to represent the number of hours it takes for Paulo and Irina collectively to paint this room. This means that the rate of them together is \(\frac{1\text{ room}}{x \text{ hours}}\). Let's fill that in, as well!
 

We can assume that, if Paulo and Irina work together at their individual rates, then the rate will be the sum. 

Therefore, \(\frac{1}{8}+\frac{1}{9}=\frac{1}{x}\). Let's solve this!
 

Therefore, the correct answer choice is the last one (which is really just a manipulation of the equation I created).

a)

If we suppose that c varies inversely with d, then this is what that statement is telling you:

• If d increases, then c decreases.
• If d decreases, then c increases.

The most basic example of this occurence is \(y=\frac{1}{x}\). As you think of larger values of x, the input, then the output, y, will decrease. 

There is only one difference from the previous function to this particular problem: We do not know the rate in which the inverse relation occurs. For example, if the numerator of the previous function was a 2, then the rate of the inverse function would double. Let's name the rate of the inverse relation "k." We, therefore, make the following equation.

\(c=\frac{k}{d}\)

b)

We can use the previous equation to figure out the rest. It involves basic substitution.

Now, let's find d when c=68:

Well, we may not need that nice diagram to answer that question!

The fourth vertex of the quadrilateral is located at \((-2,-7)\), which is the same y-coordinate as the center of dilation, \((2,-7)\). Let's assume that the center of dilation is actually the origin of the coordinate plane. 

This means that, from the original origin, we have effectively shifted the entire coordinate plane 2 units to the right and 7 units downward. Since the coordinates of the fourth vertex lie on the same y-coordinate, it is much easier to notice that the distance from the vertex to the "new origin" is 4 units. This means that the center of dilation is at \((0,0)\) and that the fourth vertex is located at \((-4,0)\)

If the question asks for a dilation by a scale factor of 3, then the distance of the vertex from our "new origin" also triples. The distance, therefore, would be 12, so the coordinate of the dilated point would be \((-12,0)\).

Of course, we must account for the fact that we shifted the Cartesian plane for convenience sake. In order to reverse these effects, we must move the plane 2 units to the left and 7 units upward, so the coordinates of the fourth vertex would become \((-10,-7)\)