Simple Factoring of Quadratics

How do you factor quadratics?

We are going to learn a simple method for factoring quadratics when the leading multiplier is 1 and the quadratic has non-complex integer solutions that are positive.
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Lets expand (x+8)(x+21)
(x+8)(x+21)=x²+29x+168

Now how would we factor something like this if we did not know what the factors are.
If we expand a quadratic, it looks like this:  (x+A)(x+B)=x²+(A+B)x+AB
The second term or the number before x is the addition of 2 roots or A+B
The last term is the multiplication of 2 roots or AB.

First If we factor 168 into all of its roots, we will find all potential factors:

168 factors into these pairs:     1         168
                                                 2         84
                                                 3         56
                                                 4         42
                                                 6         28
                                                 8         21
                                                 12       14
                                                 24         7

Now we see which 2 pairs add up to 29.  The only 2 that add up to 29 are 8 and 21.

Lets do a couple of simpler problems where we do not already know the factors:

Example 1:      x²+12x+32

Lets factor 32:                                                     32
                                                                          2    16
                                                                                2    8
                                                                                     2   4
                                                                                         2   2
                                                      or
                                                    1         32
                                                    2         16
                                                    4          8

So
1 and 32 add up to 33
2 and 16 add up to 18
4 and 8 add up to 12 so this is our solution set so it looks like this factored:
x²+12x+32=(x+4)(x+8)

One more example:
x²+10x+21

 Lets factor 21:                                                 1        21
                                                                         3         7

We see the only 2 number that add up to 10 are 3 and 7 so:
x²+10x+21=(x+3)(x+7)

In a future post we will do numbers with negative roots.

Multiplication of 2, 4, 8 

I have helped students who are given a 10 x 10 multiplication table and in school they are required to memorize it.  Many students have a difficult time with this task.  They also use speed method like using flash cards in class.  There are also classes that use the flash cards and use speed methods to learn.  The problem with speed methods is that only the best students benefit from this and the rest of the students are left behind. Some students respond well to these methods but there are many students who don't respond well to this form of learning how to multiply.  These students are left behind and many never recover from this in developing their math skills.  Here is a method I use to teach multiplication of 2, 4 and 8.

Here is a easy way to memorize the numbers:

• When I was young we used to learn numbers like 2's by rhythmically saying.  This was common on kids shows like Sesame Street.

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 

70 72 74 76 78 80

• It is very easy to memorize.  After these numbers are memorized, the student can say: 2 times 2 is 4, 2 times 3 is 6, 2 times 8 is 16,  2 times 9 is 18, and 2 times 10 is 20.The next thing is for a person the rhythmic saying and silently say 2 and then say 4 out loud and then silently say 6 and then say 8 out loud.  This can be done while looking at the chart and saying out loud the high lighted numbers.

2  4  6  8  10  12  14  16  18  20  22  24  26  28  30  32  34  36  38  40  42  44  46 48 50  52  54  56 58  60  62  64    66  68   70  72  74  76  78 80

Then a person can rhythmically say 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80.

After the numbers are memorized say: 4 times 2 is 8, 4 times 3 is 12, 4 times 4 is 16, 4 times 5 is 20, 4 times 6 is 24, 4 times 7 is 28, 4 times 8 is 32, 4 times 9 is 36, and 4 times 10 is 40.

• The last step is to rhythmically say 8 16 24 32 40 48 56 64 72 80.   A person can then say 8 times 2 is 16, 8 times 3 is 24, 8 times 4 is 32, 8 times 5 is 40, 8 times 6 is 48, 8 times 7 is 56, 8 times 8 is 64, 8 times 9 is 72, and 8 times 10 is 80.

I would recommend that a student go through the first step this 5 times twice a day for a week. Then a student can do the second step for 5 times twice a day for 3 days.  Then the student can do the 3rd step 5 times twice a day for 3 days.  If a student counts the 1's there are 35 memorized and over half way done.

	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

After the student has memorized the numbers, they can fill out their table and see their progress.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 4 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 215 120 125 130 135 140 145 150 6 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126 132 238 144 150 156 162 168 174 180 7 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140 147 154 261 168 175 182 189 196 203 210 8 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 168 176 285 192 200 208 216 224 232 240 9 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180 189 208 308 216 225 234 243 252 261 270 10 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300

Fast Math Squaring Numbers Ending in 1

There are a couple of real nifty ways of doing fast math with numbers that end in 1.  We are going to present a addition method.  There are other methods that we will present in subsequent posts but this method presented is the easiest way to calculate squares ending in 1.  

Addition method:
Lets start with a number 71.

• We will subtract 1 from this number and call it our base number: 71-1=70
• We will square our base number: 70²=4900
• We will multiply our base number by 2: 70x2=140
• We will add the square of our base number to our base number multiplied by 20 and add 1.  4900+140+1=5041

That is pretty easy isn't it?

Lets find 101²
101-1=100

100²=10,000
2X100=200
10,000+200+1=10,201 

That is incredibly easy and fast!  Lets try one more number:

191²

191-1=190
190²=36100

2X190=380
36100+380+1=36481

Outstanding!  Now you know how to square numbers ending in one and you can do it very fast.  We will show how to do other numbers in subsequent posts so stay tuned.

In the Prime of Its Life

The first primes less than 100  and rules to easily find

Here is a common scenario:

You are taking the SAT test and you are asked how many prime numbers there are between 30 and 80.  You know what prime numbers are and you know what prime numbers are but you have limited time and no table to help you.  You decide to skip the question because you do not have enough time to finish it by the end of the test.

This is a very common test question yet it is not trivial because most people have not memorized the prime numbers so here is a quick was to determine the prime numbers. 

1   2   3   4   5   6   7   8   9   10            (4 primes)

11 12 13 14 15 16 17 18 19 20            (4 primes)

21 22 23 24 25 26 27 28 29 30            (2 primes)

31 32 33 34 35 36 37 38 39 40            (2 primes)

41 42 43 44 45 46 47 48 49 50            (3 primes)

51 52 53 54 55 56 57 58 59 60            (2 primes)

61 62 63 64 65 66 67 68 69 70            (2 primes)

71 72 73 74 75 76 77 78 79 80            (3 primes)

81 82 83 84 85 86 87 88 89 90            (2 primes)         

91 92 93 94 95 96 97 98 99 100          (1 prime)

                                                 
 The sieve of Eratosthenesis is a methodology for finding all prime numbers.  Without going into how the sieve of Eratosthenes works, I will say the methodology I present will be similar but not the same and it will be easier to use for numbers between 1 and 100.  

After 2, all even numbers are not prime.  
After 5, all numbers that end in 5 are not prime.
No numbers that end in 0 are prime.

Now here is the trick.  If you are looking at the columns where the numbers end in a 1, 3, 7, or 9, you will see there are quite a few primes.  Of these numbers that end in 1, 3, 7, and 9, all numbers except 1, 3, 49, 77, 91, are divisible by 3.

Okay lets check this out.
21÷3=21              33÷3=11               27÷3=9               39÷3=13
51÷3=14              63÷3=21              57÷3=19              81÷3=27
81÷3=27              93÷3=31              87÷3=29              69÷3=23 

You can easily find any prime on numbers less than 100.  If you are asked if 87 is prime, you will say no it isn't because it is divisible by 3.  If you use the 3 divisibility rule, it is even easier.  

Also one more trick for those taking the SAT or any standardized test, the number of primes in each row after the tens are:
 2, 2, 3, 2, 2, 3, 2, 2, 1 and the ones and 10's each have 4.  Refer to the table above.
If you are asked the question, "How many prime numbers are there between 30 and 80?"  You can answer 16.  2+3+2+2+3+2+2=16