Chapter 5, Section 9 - University of Houston



Chapter 5, Section 9

Solving Equations by Factoring.

We’ll be looking at equations and solving them for x.

In the Big Picture sense, we’ll be solving for the x axis intercepts – here’s why:

Given [pic].

This is function: it’s graph passes the vertical line test…and it’s graph hits the y axis once and the x axis twice. Let’s look at why.

The y axis intercept occurs when x = 0. Calculate f ( 0) and you get 3…so the point

( 0, 3) is on the graph.

The x axis intercept happens when y = 0…which is the same a saying f (x) = 0. So for our equation, then,

[pic]. We want to find the x value when y = f (x) = 0.

This isn’t obvious from the added form of the function but it is fairly obvious from the factored form of the function.

Last time, we found out that this trinomial factors to (3x + 1 ) ( 2x + 3).

This means that asking to solve this problem for x

[pic]

is the same as asking to solve 0 = (3x + 1 ) ( 2x + 3).

Now note, on the LHS I have two multiplied numbers. When two multiplied numbers are zero it’s a sure bet that one of the numbers is zero, maybe both of them.

(3x + 1 ) ( 2x + 3) = 0 means that

(3x + 1 ) = 0 or ( 2x + 3) = 0

If we solve the first little problem for x we get x = [pic] . The second says x = [pic].

If we check these x’s in the original functional statement we find that this is true.

This gives us two more graph points. [pic]. This is true and useful and beyond the scope of this class.

Now, in this class we’re not going to focus on the function, not the graph points. We’re going to focus on getting the x’s. Your instructor in College Algebra will assume you can get the x’s and will work with you on what they mean.

So let’s solve some problems.

“For [pic], find the x axis intercepts.” This is a College Algebra question. In this class, we’ll say

“[pic] Solve for x. “

[but keep in your mind where you’re going with this]

How to do this:

The first step is to make sure the RHS is zero.

The second step is to factor out the GCF.

The third step is to factor the LHS.

Then set each factor to zero and solve for the x’s.

Report your answer.

Check your answer.

So. [pic]

The RHS is zero.

The GCF is 1.

The third step is to factor the LHS.

a = 1 here, so we’ll use Technique A – find two numbers that add to (1 and multiply to

( 2. Put them in the factors.

( x ( 2 ) ( x + 1) = 0

Then set each factor to zero and solve for the x’s.

( x ( 2) = 0 or ( x + 1 ) = 0

x = 2 or x = ( 1

Report your answer: { (1, 2}

Check your answer.

Let’s do another one: [pic]

Solving trinomial equations by factoring:

The first step is to make sure the RHS is zero.

The second step is to factor out the GCF.

The third step is to factor the LHS.

Then set each factor to zero and solve for the x’s.

Report your answer.

Check your answer.

The first step is to make sure the RHS is zero.

[pic]

It’s not. Rewrite the problem:____________________________________

The second step is to factor out the GCF: good news, the GCF is 1

The third step is to factor the LHS.

Here, a ( 1. Grab that worksheet and get this thing factored! Technique B.

Then set each factor to zero and solve for the x’s. (think: when two numbers multiply to zero, then one number or the other number is zero)

Report your answer

Check your answer – in the factored form if you’re in a hurry or, better, the original form.

Another: [pic]

Solving trinomial equations by factoring:

The first step is to make sure the RHS is zero.

It’s not. Rewrite it:

The second step is to factor out the GCF.

Good news; it’s one.

The third step is to factor the LHS. DIFFERENCE OF TWO SQUARES!

Then set each factor to zero and solve for the x’s.

Report your answer.

Check your answer

Another: [pic]

Solving trinomial equations by factoring:

The first step is to make sure the RHS is zero.

rewrite

The second step is to factor out the GCF.

1

The third step is to factor the LHS. ( a ( 1, Technique B)

Then set each factor to zero and solve for the x’s.

Report your answer.

Check your answer.

Here’s one that will require some initial cleverness:

[pic]

Solving trinomial equations by factoring:

The first step is to make sure the RHS is zero.

This looks pretty awful til you recognize that you can multiply both sides by 75 and clear the fraction away.

[pic]

This isn’t so bad. Rewrite:

The second step is to factor out the GCF.

The third step is to factor the LHS.

Technique B

Then set each factor to zero and solve for the x’s.

Report your answer.

Check your answer.

Note that is a popular type question, it is in your homework, on your quiz, and on your worksheet…you should be getting a real strong hint, here, people…

Looking at [pic]

The first step is to make sure the RHS is zero. It is

The second step is to factor out the GCF.

The third step is to factor the LHS.

Make sure you don’t lose track of the GCF it has an x in it and is one of your factors!

Then set each of the 3 factors to zero and solve for the x’s.

Report your answer.

Check your answer.

DO NOT DIVIDE THROUGH BY X. EVEN IF THE PROBLEM IS WRITTEN LIKE

[pic]’

The most you can divide out is a 2. If you divide by x, you divided by the answer x = 0 and dividing by zero will get you put in Math Jail.

Here’s another one with a real live GCF:

[pic]

The first step is to make sure the RHS is zero.

rewrite:

The second step is to factor out the GCF.

Hold on to the GCF; it is one of your factors!

The third step is to factor the LHS. (Technique A)

Then set each factor to zero and solve for the x’s.

Report your answer.

Check your answer.

Same warning: Don’t get all happy thinking you can divide both sides by x and have an equivalent equation. You divide out the solution x = 0 and you lose all the points. You can only divide when you know the number is NOT 0…you don’t know this about an x.

For [pic]

The first step is to make sure the RHS is zero. check

The second step is to factor out the GCF.

The third step is to factor the LHS.

It’s already factored. When c = 0, you have a LOT less work to do. Look for these in your homework. They’re not meant to be tricky; they’re meant to be a “gimme”. The alternate form of this question is[pic].

Then set each factor to zero and solve for the x’s.

Report your answer.

Check your answer.

Worksheet for solving trinomial equations

Solve: [pic] for all natural numbers x.

The first step is to make sure the RHS is zero.

The second step is to factor out the GCF.

The third step is to factor the LHS.

Then set each factor to zero and solve for the x’s.

Report your answer.

Check your answer.

When you’re done working your answer is [pic] since neither of these is a natural number, the answer to the question above is “no solution”. There are several of these on your weekly quiz. Read the question carefully!

Worksheet for solving trinomial equations

Find all natural number solutions to [pic]

The first step is to make sure the RHS is zero. check

The second step is to factor out the GCF. is one

The third step is to factor the LHS.

Technique B!

Then set each factor to zero and solve for the x’s.

Report your answer.

you got x = [pic] or x = 3. Only x = 3 is a natural number so The Answer is x = 3.

Check your answer.

Find all whole number solutions to [pic]

The first step is to make sure the RHS is zero.

rewrite:

The second step is to factor out the GCF.

There is one … keep track of it!

The third step is to factor the LHS.

no need … after you take out the GCF you’ve got a linear term

Then set each factor to zero and solve for the x’s.

Report your answer.

You get x = 0 or x = (5. (5 is not a whole number; it is purely an integer.

So x = 0 is The Answer.

Check your answer.

Worksheet for solving trinomial equations

The first step is to make sure the RHS is zero.

The second step is to factor out the GCF.

The third step is to factor the LHS.

Then set each factor to zero and solve for the x’s.

Report your answer.

Check your answer.

Worksheet for solving trinomial equations

The first step is to make sure the RHS is zero.

The second step is to factor out the GCF.

The third step is to factor the LHS.

Then set each factor to zero and solve for the x’s.

Report your answer.

Check your answer.

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download