Happy Factorizing!

Here are some more examples of the factorization method below. Take a look!

10y² - 23y +12

(2y - 3)(5y - 4) 

Check in and out:  -15y (inner) - 8y (outer) = -23y (center term)

CORRECT.

9m² + 18m - 7

(3m - 1)(3m +7)

Check in and out: -3m (inner) + 21m (outer) = 18m (center term)

CORRECT AGAIN.

Wow, I'm on a roll. I could do this all day, and I would too, if wasn't for school and sleep and homework and... stuff.

Factorisation Rules!!

Fully factorize the trinomial

This is a trinomial and the directions were to fully factorize the given trinomial. When factorizing a polynomial you must first check to see if anything is in common, if yes, take it out, if no, open two sets of parentheses.

In this problem my mistake was that I automatically assumed that it was not factorisable simply because I did not think that any factors of ten could be added or subtracted to get the "invisible" 1 in the term in the center. However it is factorisable because the 3 in the first term should and can be divided into both sets of parentheses to "help" the the factors of the last term create the inner one.

ShouldaCouldaWouldaTime... This is what I should have done.

   3k² + k - 10

(x  +  2)(3x -  5)

DON'T FORGET TO CHECK!

We check by multiplying inside terms (2 + 3k) and outside terms (-5 + k) and adding to make sure we get the inner term (1k).

(2 x 3k) + (-5 x k)

6x-5x = 1k

So the answer (x  +  2)(3x -  5) is correct.

And that's my answer! I really isn't that hard, especially when you have FULLY MASTERED THE CONCEPT *hint hint* like me... :)

Want more proof on the fact that I know what are and when to use All Real Numbers and No Solution? Check these out!

7x - (3x - 4) = 7x + 9

7x - 3x + 4 = 7x +9

4x + 4 = 7x + 9

NO SOLUTION.

9x ( 3x - 1) = 3 (9x² - 3x)

 27x² - 9x = 27x² - 9x 

                                              + 9x           +9x 

27x² = 27x² 

ALL REAL NUMBERS. 

 Never should you doubt my skills. I shall simply prove you wrong.  :)

No solution? Undefined? All real numbers? Peice of cake.

A) Create and solve an equation that has no solution. B) Explain (clearly) when an equations has no solution

This was a question from the Equations and Inequalities POK I took on October 8th, back when I had a hazy understanding of "No Solution", "All Real Numbers", and "Undefined".

My mistake in the above problems was that I confused the definitions of "No Solution" and "All Real Numbers" in my creation of a "No Solution" problem and my definition of an equation with no solution was inaccurate and ambiguous.

Well, after some clarifications with my teacher and my notes I can now say I know what each of those terms mean. Here are my corrections.

NO SOLUTION: When the solution obtained is not true and no value of 'x' will satisfy the equation.
Example:
2+7x=6+7x
  -7x      -7x
     2=6
2 does not  equal 6 so the equations has no solution.

UNDEFINED: When the solution has zero in the denominator.

ALL REAL NUMBERS: When any and all real numbers substituted in for 'x' will satisfy the equation.
Example:
6+9x=6+3(3x)
6+9x=6+9x
-6     -6
    9x=9x
This sentence is true therefore it will be true for all real values of 'x'.

I threw in some extra definitions, just so the differences between them would be clear. Ahh the joy of learning.

A mistake made is a lesson learned.

So to show you how I've learned from my mistake I'll create a sample problem that is similar to the one below.


The width of a rectangle is 2 meters less than twice it's length. The perimeter is 32 meters. Find the dimensions of the rectangle.


What we know: Width (w) = 2l - 2
                          Perimeter  (p) = 32 m

What we don't know: Length ( l )

First, find the length:   
 p = 2w + 2l

32 = 2(2l - 2) + 2l
32 = 4l - 4 + 2l 
32 = 6l - 4
+4        +4
36 = 6l
/6     /6
6 =  l (The length of the rectangle is 6m)

Now, find the width:
w = 2l - 2

w = 2(6) - 2
w = 12 - 2
w = 10 (The width of the rectangle is 10m)

Then, check!!
p = 2w + 2l

32 = 2(6) + 2(10)
32 = 12 + 20
32 = 32 (Therefore the answer is correct)

My final answer (in a full sentence) : 

The dimensions of the rectangle are 6 x 10 m.

Booyah. :)

Oops, my mistake... I mean, mistakes.

The length of a rectangle is 6 meters less that twice it's width. The perimeter is 48 meters. find the dimensions of the rectangle.

I got this lovely multi-step word problem wrong on my math Common Assessment on October 14th 2010.  I spoted some mistakes as you can see above. I...

In finding the width:
1) Didn't distribute the 2 to EVERYTHING in the parentheses. I should have gotten 2w+4w-12
2) After doing that I added 48 and 6 and got 42. What was I thinking? The world may never know. I should have added 12 to 48 and have gotten 60.

In finding the length:
1) I substituted 7 in for 'W' but 7 did not satisfy the equation 2w+2(2w-6)=48. Therefore my answer will automatically be wrong.  

But never fear, I do it correctly here!

and here!

To check we substitute the "w" and the "l" in an equation that translates to the perimeter of the rectangle.

P=(2l)(2w)

And so...

48=2(14)+2(10)

48=28+20

48=48

Therefore the correct dimensions are 14x10

Not too shabby...

It's great to be able to write a correct answer in bold letters, it means you know what you're doing. Talk about smarticle particles.