GED Question from Austin: Quadratic Equations and Graphs

Whoa. Austin’s really into the hard stuff. Here’s his question:

hi im austin.
i have a question on some things. im having trouble with the quadratic equations and graphs. the quadratic equation i just cant seem to understand quite that well. The graphs…well i just cant figure out how to solve the slop of a line and everything else. You think you could maybe give me some good advice on an easy way to understand everything about quadratic equations and x and y axis graphs.

Part 1: Quadratic Equations for the GED

Yeah, this is the hard stuff. Quadratic equations… the good news is, you won’t have a lot of them on the GED test. The first thing is, what is a quadratic equation, anyway? It’s an equation with x squared in it, basically. Usually, it has something x squared, plus or minus something x, plus or minus a number, equals something. Like: x² + 3x – 4 = 0

Quadratic equations are hard to solve. I’m just gonna deal with the easier ones, those’ll be the kind on the GED. That means, there’s no number before the x squared. When you get an answer, there’s gonna be two possible numbers for x, because of the square. That’s because a square root can be negative or positive. So -1 squared is 1, and 1 squared is also 1.

There’s a big, confusing formula for finding x in a quadratic equation like this one. But there’s also a shortcut. The best way to solve these (the simple ones, that is, which are the ones on the GED), is to factor. That just means, find two things that multiply together to equal the equation. To do this, you’ve kind of got to work backwards. Your factors are going to look like this: (x + ?) (x + !) = 0. Remember, the question mark and exclamation mark can be negative or positive numbers.

If you multiply out (x + ?) and (x + !) you get x² + (! + ?)x + (! times ?).

It don’t matter what the numbers are… the number by the x is going to be your two numbers added together, and the number without the x is going to be the two numbers multiplied together. So, if the example is this: x² + 3x – 4 = 0, what two numbers can multiply together to get -4? (You take the minus sign and put it with the 4. That’s the big stumbling block…) You can multiply 4 times -1, or -4 times 1, or 2 times -2.

So, which of those, added together equals the middle number, +3? It’s 4 and -1. So, x² + 3x – 4 = (x + 4) (x + -1)

So, (x + 4) (x + -1) = 0. That means, either x + 4 or x + -1 will equal 0. The possible answers for x are -4 and 1, the opposite signs of your numbers. Did you follow all that?

On the GED, it can be easier to try the answer choices in the equation, to see which ones are right. For more about quadratic equations, check out:

http://www.themathpage.com/alg/quadratic-equations.htm

http://www.purplemath.com/modules/quadform.htm

http://plus.maths.org/issue29/features/quadratic/index-gifd.html

Part 2: Graphs for the GED

x and y axis graphs… well, here’s the deal. Any kind of equation with two variables (an x and a y) can be graphed as a line. Lines’ll have different shapes, depending on the equation. You’re really askin’ about graphs of lines. That means, there’s no squares or anything funny in the equation. There’s an x and a y, and the line is made up of all the possible numbers for x and y that can make the equation true.

For x = y, the numbers for x and y will always be the same. So, if x =1, then y = 1. If x = 2, y = 2. The line will be a straight line on a graph where all the x’s equal all the y’s. Okay, it’s easier to look at it. Go here to get a great intro to graphing lines:

http://library.thinkquest.org/20991/prealg/graph.html

So, you asked about slopes. That means, how steep is the line on the graph, and what direction does it go? So, the slope’s got two parts, and the answer to those questions is put into a number.

First, how steep is the line? You can figure it out by picking two points on the line, and seeing how far UP and then how far OVER you’ve gotta go to get from the first point to the second. Divide how far you went UP by how far you went OVER. That gives you a number.

Then, which direction is it going? Going from bottom to top, does it lean to the right? Then the slope’s a positive number. Does it lean to the left? Then the slope’s a negative number. Here’s a link you can go to to practice finding the slopes of some lines.

Slope Practice Worksheet: http://www.algebrahelp.com/worksheets/view/graphing/slope.quiz

Also, check out more about graphing here: http://library.thinkquest.org/20991/alg2/graphs.html

You can also figure out the slope just from an equation, if it’s said in a particular way, called Slope Intercept form. Slope Intercept form has the number for the slope in it, and the number for the y-intercept (where the line passes through the y axis, or where x=0). It goes like this: y = mx + b

m = slope, and b = y-intercept

So, if your equation is y = 2x + 3, then the slope is 2. Makes it easy to find, huh?

Here’s a cool page lettin’ you see how the Slope Intercept form looks like as a line:

http://id.mind.net/~zona/mmts/functionInstitute/linearFunctions/lsif.html

Here’s some practice making lines using Slope Intercept form:

http://www.algebrahelp.com/worksheets/view/graphing/slopeintercept.quiz

For more information about the GED test and GED test preparation, visit The GED Academy at http://www.passged.com.