Word Problems With Quadratics

Thursday

On Thursday, 4/4, we learnt about solving word problems that would require us to use the quadratic formula. To refresh your memory, if it's not already memorized, here is the quadratic formula:

At the beginning of class we split off into groups that we would later make notability projects with. Before that we did the homework problems up on the board.

This is the problem that my group worked on:

A deck of uniform width has area 72m^2 and surrounds a swimming pool that is 8m wide and 10m long. Find the width of the deck.

It always help to draw out the problem to figure out how to solve it.

Above are the steps we went through to solve the problem using the equation "length x width = area". Our length is 8 + 2x because we know that the pool is 8 m wide but there are also two areas of x on both side of the pool. The width is 10 + 2x for the same reason. When we multiply these two we get 72 + 80 because the problem says the deck's area is 72 and we can find the area of the pool by multiplying 8 and 10. After simplifying the problem we get an equation that we plug into the quadratic equation and simplify until we get x = (-9 +/- 3sqrt17)/2, since lengths can be negative, we know that the sign before the root is +, not -. We plugged this into our calculator and got close to 5/3.

After doing these problems we sat down with our groups and started working on more word problems.

Friday

On Friday we finished on our group word problems. Here is the problem that my group worked on, shown in a notability note:

This problem was solved much like the example I showed before, we set up the problem much like any other word problem, trying to solve the x variable, and then when we got to the quadratic we plugged it into the quadratic formula and simplified some more. Since this is the blog you can't hear the recording that is in our project but if you want to hear it, or see any of the other projects from other groups I would recommend going and checking out the dropbox where the all are.

Thanks for reading, hope you learnt a lot!

-Rowan

The Quadratic Formula and Imaginary Numbers

In the past two days of class, we touched on two major topics:

-How to apply the Quadratic Formula to quadratic equations, along with the parabolas that accompany them (along with the memorization of the formula);
-Imaginary numbers, what they are, what they mean and how to apply them in mathematics.

So, to dive right in--

The Quadratic Formula
As I'm sure you have heard before, the Quadratic Formula is a formula that allows you to plug in values "a", "b", and "c" from an equation, solve it, and from there obtain your x values.

Here it is:

This equation is used to find the value of x in a quadratic example.

a=coefficient of x^2
b=coefficient of middle term x
c=third term

You simply plug these values in for a particular equation, solve it, and out comes your (usually) two x values.

An Example:

The first step of this is to determine the value of y. Say y is zero, and the rest is easy. Here are the values for a, b, and c:

a=1

b=-10

c=9

Then, you plug these values into the quadratic equation (above). This comes out to:

x=(10 +/- sqrt[100-36])/2

x= (10+/- 8)/2

x= 9, 1

I hope this quick overview of the Quadratic Formula has helped in solidifying your understanding. 

Quadratics in Parabolas

Every quadratic equation, when graphed, comes out to a parabola; meaning that it resembles some kind of a "half-pipe" in its line. There are three kinds of parabolas in quadratics (disregarding sideways vs vertical), displayed below:

Let's take a look at the graph of the quadratic solved previously in the post;

As you can see, the quadratic x^2-10x+9 is a parabola as it resembles a half-pipe. If you take a look at where the line intersects the x-axis, you'll see that those points are the two solutions of the equation. This applies to all quadratics looking for an x value.

The number of solutions a quadratic has (one, two, or zero) is based on the discriminant in the Quadratic Formula. This is the b^2-4ac, of course with the a, b, and c values plugged in the with the relative equation.

Here is a chart developed in class that distinguishes between one, two, or zero solutions in a quadratic:

Note that the "one x-intercept (solution)" side includes two intercepts as well. No x-intercepts include negative square roots.

Imaginary Numbers
Contrary to its name (and the several puns that accompany it; I'll spare them here, for the most part) imaginary numbers are very much real.

They are represented by i; an abbreviation for a definition that is actually very simple:

i=sqrt(-1).

Yup, that's it.

Here is a list that we brainstormed in class about imaginary numbers and what they mean (after a quick internet source on our fancy imaginary--I mean i--Pads).

Here is a diagram with the explanation of what "complex numbers" really are (we didn't seem to understand it at first):

And some quick work with i and its exponents, following a remarkable pattern:
i=i
i^2=-1
i^3=-i
i^4=1
i^5=i

I hope this post has helped your understanding of Quadratics, Parabolas, and our newly-discovered topic of imaginary numbers.

--Peter

Solving Radical Equations

Hello Class,
Sorry for the late post but here is all the information for March 27th 2013 and March 28th.  I will show you some step by step examples:


Check out this video for more information on Sums of Radicals
http://m.youtube.com/watch?v=ZA497RyAKe8

Here is the homework or practice problems that you can try out :)

Page 259: problems, 1-25 (every other odd)

And page 261: problems, 1-25 (every other odd)

Hre is the class information to Solving Radical
 equations. Here is a video to start you out. :)
http://m.youtube.com/watch?v=LY8VBsLf-4M

Here is the homework or practice problems that you can try out :)

Page 261: problems, 1-25 (every other odd)

Page 259: problems, 1-25 (every other odd)