DISTRIBUTIVE PROPERTY (CONTINUED)
REVIEW OF DISTRIBUTION:
Let’s say we are given:
-(x-6) = 13
Well, we know there is a negative outside of the parenthesis, so we need to distribute the negative.
We do that by multiplying each term by the negative.
So:
-(x - 6) = 13
So:
-1 * x = -x
And
-1 * -6 = +6 (or just 6)
So now we are left with:
-x + 6 = 13
- 6 - 6
-x = 7
x = -7
DON’T FORGET, THIS IS THE EASIEST EXAMPLE THOUGH
Again, it’s still important to know how to distribute a number to an equation, however just distributing one element (number or variable) is still pretty simple.
Of course, as we saw, the level of difficulty gets higher when you start dealing with variables as well.
But before that, let’s do another one really quick with an actual number this time.
EASY EXAMPLE #2:
-3(2x - 15) = 75
Again, we know that -3 is being multiplied by the equation.
So let’s distribute that -3:
-3(2x - 15) = 75
So:
-3 * 2x = -3 * 2 * x = -6 * x = -6x
-3 * -15 = 45
Now we have:
-6x + 45 = 75
- 45 - 45
-6x = 30
-6 -6
x = -5
So now we get to the next level
Again, simply multiplying a number by an equation is the easiest way to multiply it out.
We know it’s much harder to multiply an equation by a variable instead of a number.
So, let’s review really quick over what happens when we are presented an equation being multiplied by a variable.
The harder example
Again, looks tough
x = +4, -4
Finally, the most difficult type
This is the only time when you use the acronym F.O.I.L., when you are multiplying two linear expressions by each other.
(Mainly because anything less is not 4 steps, and anything more is still not 4 steps. So this is pretty much only for 4 stepped equation, or another way to say this is a quadratic)
The F.O.I.L. method
So, let’s say we have something like:
(-4x + 40)(x + 10) = 0
This is where the F.O.I.L. method can be used.
Again, remember F.O.I.L. stands for:
First
Outer
Inner
Last
It’s an organizational acronym to help you multiply what you need to without over doing it.
EXAMPLE OF THE F.O.I.L. METHOD
Now we put everything together
x = +10, -10
Example 2
Solving with what we have:
SOLVING USING THE BOX METHOD
So we know how to distribute using F.O.I.L., but are there any other easy ways to distribute?
Actually, there is; it’s called the box method.
What you do with the box method is, you put each element in an equation on the top of the box, then you put the other elements to the other equation to the side.
Then you multiply them together and come up with your answer.
(Again, we’ve reached that level in math where it’s easier to show than explain.)
Example 1:�
Let’s use the equations that we know the answer to as an example.
So, let’s say we have:
(-4x + 40)(x + 10) = 0
If we want to use the box method, here’s how we would do it:
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First we make our box (always make sure to add an extra row and column to your boxes. So in this example, we have 2 elements by 2 elements, so we need a 3 by 3 box.)
Next, pick an equation and put the elements on the top of the equation.
Then put the elements of the next equation on the side of the box.
Now multiply each part individually!
Now add what you have together.
-4x
+ 40
x
+ 10
40x
-40x
400
Now we solve with what we have!
Example 2 of the box method:�
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First we make our box (always make sure to add an extra row and column to your boxes. So in this example, we have 2 elements by 2 elements, so we need a 3 by 3 box.)
Next, pick an equation and put the elements on the top of the equation.
Then put the elements of the next equation on the side of the box.
Now multiply each part individually!
Now add what you have together.
x
- 6
x
- 7
-6x
-7x
42
Solving with what we have:
SO, EITHER WAY WORKS
Like I’ve said time and time again, there’s usually at least 2 or more ways to solve a single problem.
So why are you being taught this?
Because, for some of you F.O.I.L. is the easiest way to solve these equations.
For others of you, the box method is the easiest way to solve these equations.
For the rest of you, maybe you saw these two ways and found a way that works specifically for you.
Either way, use what you think is the best way, because the reality is, that’s the way you’re going to solve these equations in the real world.
LASTLY, REMEMBER, ZERO IS YOUR FRIEND HERE.
Now try some on your own!
SO WHAT HAVE WE LEARNED?