Questions Week 3: Math112

*Take any number (except for 1). Square that number and then subtract one. Divide by one less than your original number. Now subtract your original number. Did you reached 1 for an answer? You should have. How does this number game work? (Hint: Redo the number game using a variable instead of an actual number and rewrite the problem as one rational expression). How did the number game use the skill of simplifying rational expressions? Create your own number game using the rules of algebra and post it for your classmates to solve. Be sure to think about values that may not work. State whether your number game uses the skill of simplifying rational expressions.

Explain Details:

Solving that first game:

x^2-1

(x^2-1)/(x-1)

(x^2-1)/(x-1) – x

Expand this:

(x+1)(x-1)/(x-1) – x

Cancel out the x-1:

X+1 – x

= 1

Here is a game:

Here is a number game that uses the skills of simplifying rational expressions.

Take any number (except for -4) and add 2. Next, multiply by 2 less than the number. Add 3 times the original number. Divide by 4 more than the original number. Finally, add 1. You should be back where you started!

Here it is with symbols:

Take any number (except for -4) and add 2.

x
x+2

Next, multiply by 2 less than the number.
(x+2)(x-2) = x^2 - 4

Add 3 times the original number.

x^2 + 3x - 4 = (x+4)(x-1)

Divide by 4 more than the original number.
(x+4)(x-1)/(x+4) = x-1

Finally, add 1. You should be back where you started!
x

*How is doing operations (adding, subtracting, multiplying, and dividing) with rational expressions similar to or different from doing operations with fractions? Can understanding how to work with one kind of problem help understand how to work another type? When might you use this skill in real life?

Explain:

Doing operations on rational expressions is very similar to doing operations on fractions. You first need to come to a common denominator for addition and subtraction. Then you can perform the operations on the numerators. For multiplication, the process is similar to fractions, as well: you multiply the numerators and denominators separately. For division, you flip the second rational expression and then multiply.

4. When solving a rational equation, why is it necessary to perform a check?

Explain in at least 50 words:

There are a few reasons to check your answers. For example, you may have introduced "false" solutions while solving the equation. Also, the solution you came up with might make the original equation undefined (if it makes the denominator of a fraction in the original equation equal to 0, for example).

*Why is it important to simplify radical expressions before addingor subtracting? How is adding radical expressions similar to adding polynomial expressions? Howis it different? Provide a radical expression for your classmates to simplify With answer.

Explain:

You cannot directly add terms inside a radical, so you need to try to get the number inside the radical to be the same in each term. It is similar to simplifying polynomials because you can only add or subtract like terms. It is different because you don’t have powers of x.

For example:

sqrt(25) + sqrt(4)

Here is the solution:

You can't just add the 25 and 4. Instead you simplify each one first:

5 + 2

= 7

*Describe two laws of exponents and provide an example illustrating each law. Explain how to simplify your expression. How do the laws work with rational exponents? Provide the class with a third expression to simplify that includes rational (fractional) exponents.

Explain:

Multiplication of like bases. When you multiply two exponential terms with the same base, you add the exponent. The formula is m^a * m^b = m^(a+b).
Example: 4^4 * 4^2 = 4^(4+2) = 4^6 = 4096
Division of like bases. When you divide two terms with like bases, you subtract the exponents: m^a / m^b = m^(a-b).
Example: 4^8 / 4^6 = 4^(8-6) = 4^2 = 16
How do the laws work with rational exponents? Provide a third expression to simplify that includes rational (fractional) exponents
The above laws are unchanged for rational exponents...
Example of the multiplication law with rational exponents: 16^(1/3) * 16^(2/3) = 16^1 = 16

6.What is the Pythagorean theorem? How is it used?

Explain in at least 50 words:

The Pythagorean Theorem allows us to determine the length of a missing side of a right triangle, given the other two sides. It says a^2+b^2=c^2, where a and b are the lengths of the two short legs, while c is the length of the longest leg, the hypotenuse. For example, if we are given two sides as 9 and 12, we can get the hypotenuse: sqrt(9^2+12^2) = 15

*How do you know if a quadratic equation will have one, two, or no solutions? How do you find a quadratic equation if you are only given the solution? Is it possible to have different quadratic equations with the same solution? Explain. Provide your classmate’s with one or two solutions with which they must create a quadratic equation.

Explain:

If you have an equation in the form ax^2 + bx + c, you can look at the discriminant, which is the value of b^2 – 4ac. If that value is positive, there are two solutions. If it’s zero, there is one solution. If it’s negative, there are no real solutions.

To get the equation, given solutions a and b, you set up the following binomials:

(x-a)(x-b)

You can then “FOIL” these terms to get a quadratic equation in the form:

x^2 – ax – bx + ab

Yes, it is possible. Equations that are multiples of each other will have the same solutions. For example, x^2 = 0 and 2x^2 = 0 both have a single solution at x = 0.

Example:

One solution: x = -5

(the answer equation is (x+5)^2 = 0)

*Quadratic equations can be solved by graphing, using the quadratic formula, completing the square, and factoring. What are the pros and cons of each of these methods? When might each method be most appropriate? Which method do you prefer? Why?

Explain:

The graphing method allows you to visualize the solution, since the solution(s) occur where the graph crosses the x axis. If you don’t have a graph to begin with, though, it might be difficult to draw the graph based on the equation alone. You’d need to plot many points to get an accurate picture.

The quadratic formula always gives the answer in a step by step expansion of the formula. There are many steps in simplifying the formula, so it takes a little while. Completing the square is similar, in that if you follow a set of steps, you will arrive at the answer, but it can take a while.

Factoring is the quickest method if you can easily see the factors of the equation. It just takes one step to write them down. If you aren’t good at “seeing” the factors, though, it can be tricky to just come up with them.

I would use the graphing method if I am already presented with a graph of the equation. The quadratic formula and completing the square are most useful for equations with solutions that aren’t integers. Factoring is useful for “simple” equations with integer solutions.

I prefer to use the factoring method, whenever possible, since it allows me to get the solutions in only one step. Each of the other methods take longer, since you have to create an accurate graph, go through the process of completing the square, or simplify the complicated quadratic formula.

If you are looking at a graph of a quadratic equation, how do you determine where the solutions are?

Explain:

You can determine the number of solutions by counting the number of times the graph crosses the x-axis (horizontal axis). Each one of those is a place where the equation equals zero, and hence a solution of the equation. A quadratic crosses the x axis either zero, one, or two times.

Explain: How is Math and\or algebra can be useful tools in personal or professional every day life

Explain in at least 100 words:

Math is extremely useful in my everyday life, and it is all around me. This course has made me realize just how prevalent it is. For example, I am able to use the Pythagorean Theorem to quickly estimate the side lengths of right triangles I see. When I see a ladder leaning against a building, the theorem immediately comes to mind. Since taking this course, I have also started thinking about relationships between things in terms of functions. I think of the “domain” and “range” for all sorts of things, such as houses and their occupants. In the same vein, I also tend to think of invalid inputs – or values that must be excluded from these domains, such as negative house numbers.