Factoring

Another way to find these points (which most likely need to be known in order to graph the parabola is to factor the equation out. In case you’re not familiar with the term “factor the equation,” let me clarify. If one were to factor the quadratic equation x² + 3x – 4, it would look like this: (x + 4)(x – 1) = 0. How did I get that, you ask?

To factor an equation one must use the last two coefficients of the quadratic equation (b and c) and think of which two numbers can multiplied to make c and added to make b. Just remember that if your c is positive your two numbers can either both be positive or both be negative. If your c is negative one of your two numbers has to be negative.

So, if I wanted to factor our equation x ² + 3x - 4 to get which points on the x-axis its parabola crossed, I would say to myself “What two numbers are multiplied to equal -4 and added to equal 3? One of them has to be negative” and then I would go through each possible pair of numbers until I came to 4 and -1.

Example 1
Let’s try to factor something a little more complicated. Say you’re given the equation x² -17x + 72. Let’s go through some possibilities.

First, always start with two parenthesis and your xs. ( x     )( x     )

Then, it’s always easier to think of what can be multiplied to make c before asking yourself if it can be added to make b. So, we start with the basics:
1 * 72 can’t work because those would add make a much higher number than -17.
-1 * -72 can’t work because those would add to make a much lower number than -17.
2 * 36 can’t work because those would make a much higher number than -17.
-2 * -36 can’t work because those would make a much lower number than -17.
3 * 24 can’t work because those make a much higher number than -17
-3 * -24 can’t work because that makes -27.
4 * 18 can’t work because that makes a higher number than -17
-4* -18 can’t work because that makes -22.
6 * 12 can’t work because that makes 18.
-6 * -12 can’t work because that makes -18 (only one off; we must be getting closer.).
8 * 9 can’t work because that makes 17.
-8 * -9 DOES work to make -17!

So, your parenthesis will look like this:
(x – 8)(x -9) = 0

Next, just set each individual parenthesis to zero:
x – 8 = 0      x – 9 = 0
and solve to get x = 8 and 9.

Whew, that was quite a workout. The thing is, you definitely don’t have to go in order starting with 1 and a lot of the time, only after a few mistakes (much less than eleven) you will get your answer. I just wanted to show what some problems can be like when you try to factor them. After a few hit and misses, you may decide you lack the patience to try any more and just go straight to the quadratic formula, anyway.

The sad truth is that last one was not even a very complicated one. Some quadratic equations have more than 1 for their a value or have a negative value for a.

Example 2
For example, for the equation -3x² -2x +8, it would be smart to start out with your parenthesis looking like this:
(-3x   )( x   )

If you wanted to avoid even more hassle, you could even multiply the whole equation by -1 so you didn’t have to deal with keeping track of the negative 3 (just don’t forget the true equation because this parabola will open downwards and if you keep your equation with 3x², you’ll graph it opening upwards which would be incorrect):
3x² + 2x -8
(3x   )(x   )

Then you have to say to yourself, “What two numbers are multiplied to equal -8 and, after one of them is multiplied by 3, added to make 2? One of them has to be negative.” Then you have to guess and check not only which one should be multiplied by 3, but also which one is negative.

Luckily, only 1 and 8 *and* 2 and 4 can be multiplied to make 8.

I have a hunch that 2 and 4 are more likely than 1 and 8 to work, so let’s start with those.

Let’s try making 4 the one that has to be multiplied by 3 and see where that takes us:
4(3) + 2 = 14
4(3) + (-2)
12 – 2     = 10
Okay, how about -4 by 3:
-4(3) + 2 = -10
(You don’t have add -12 to -2 because only one of the two numbers (the 4 or the 2) can be negative.)

Since we didn’t get 2, 2 (or negative 2) must be the one that has to be multiplied by 3 to get our answer:

2(3) + 4 = 10
2(3) + (-4) = 2
6 -4 = 2
Okay, so 2 and -4 are what we put in the parenthesis:

(3x – 4)(x + 2) = 0

3x - 4 = 0     x + 2 = 0

3x = 4      x = -2

x = 4/3 or 1 1/3

We now know that our parabola opens downward and crosses the x-axis at 1 1/3 and -2.