2.1 Quadratic Expressions and Equations

Solving Equations

Let's say you've got a see-saw (teeter-totter)... and you've got 50 pounds of stuff piled on each side:

Here's the big Algebra game:

Whatever you do, you've got to keep the see-saw balanced!

What if we add  3 pounds  to the left side?

CLUNK!  It's not balanced anymore!

But, if we add  3 pounds  to BOTH sides?

It stays balanced!

Equations are just like see-saws...

You have to keep them balanced!

So, whatever you do to one side of the " = " you've got to do to the other side!

EXAMPLE 1:

Let's start with an easy one:

Solve  

We can just look at it and see that  x = 10...

But, what if we didn't see that?  What would we do?

Here's the Algebra trick:

We'll add 3 to both sides!

*Remember the see saw?
Whatever we do to one side of the equation,
we have to do to the other side.

Why did we ADD 3?

             To undo this!  + is the opposite of -

The goal is to get the x alone.  Just imagine that Mr. X hasn't showered in a few weeks and everyone wants to get away from him.  It's your job to help.

EXAMPLE 2:

Solve  

(Yes, I know you can see that the answer is 2, but we're learning to play a game here...  They're going to get a lot harder.)

Your mission:  Get the smelly X alone.

Who needs to get away?  The +5!

What will undo a +5?  -5!

Do it to both sides!

Here's the first thing we can do:

We can add or subtract something from both sides!

We've already learned that we can add or subtract something from both sides of an equation.

EXAMPLE 3:

We need to get this 4 out of here...

What's he doing to the x?  Multiplying!
What's the opposite of multiplying?  Dividing!

So, divide both sides by 4:

Here's what's going on with this thing:

EXAMPLE 4:
 

Solve

There are a couple different ways to deal with this one:

WAY 1:

Rewrite it like this:

Since the 4 is dividing into the x,
we'll multiply both sides by 4 to undo him:

WAY 2:

Use the fraction fact that 4 is the multiplicative inverse (big word time!) of 1/4:

Remember:

Does x = 36 work?  Check it!

Here's the way you do NOT want to do this guy!  (And do NOT let me catch you doing it!)

Do NOT divide both sides by 1/4!

EXAMPLE 5:

Solve

There are two guys bugging the x...  -7 and 2.  The 2 is really locked on and the -7 is

Now, ditch the 2:

Check it!

Solving Inequalities

For these problems, instead of just having one answer, like x = 3

We are going to get a range of answers, called an interval...  Like this:

This means that x can be 1 ... or x can be bigger than 1 ... x can be 2 or 4.671 or 1.00001 or 50,000,000.

One way to write this interval is

(read "x is greater than or equal to 1" )

So, what would this mean on a number line?

Just read it!  "x is less than 3."

We put an open dot at the 3, since x cannot be 3.

So, x < 3 means that x cannot be 3, but it can be a number less than 3... like 2.998 ... or 0 ... or -5931.

What about something like this?

This means that x can be 1... or x can be 4 ... or x can be any number between 1 and 4 ... Like 2 or 3.01459.

One way to write this interval is

(read as "x is greater than or equal to 1 and less than or equal to 4")

So, how would we graph this on a number line?

It says that x cannot be -1...

It says that x CAN be 3...

And it says that x can be a number between -1 and 3...

What about something like this?

If you have to be just ONE x guy, you either have to pick an x on the left side (in x < 0) OR an x on the right side (in x > 3).  Right?

So, we write it like this:

All the guys we've been working with are called "inequalities" since they have the symbols ,  ,  ,    instead of just  =  signs.

What about something like this?

If you have to be just ONE x guy, you either have to pick an x on the left side (in x < 0) OR an x on the right side (in x > 3).  Right?

So, we write it like this:

All the guys we've been working with are called "inequalities" since they have the symbols ,  ,  ,    instead of just  =  signs.

Absolute Values 

·                        lxl< a ⇔ −a<x<a

·                        lxl > a ⇔ x<−a, x>a