Completing the Square
For Leading Coefficients that aren't 1

We looked at completing the square for equations who's leading coefficients are 1.  If the leading factor is not 1 we factor the number out of the leading coefficient and the second term so that the leading coefficient is one and then we proceed to complete the square.

The technique is very easy.

Lets look at a quadratic equation in the standard form.

Y=AX²+BX+C 
We factor A out of both the A and B term.
Y=A(X²+[B/A]X)+C 
Notice how we don't factor the A term out of C

To write this equation in vertex form, we take 1/2 of the B/A value and write our equation like this:

Y=(X+[1/2][A/B])²+k

It looks kind of messy but lets use an example.
Y=2X²+4X-3

Our equation now looks like this:

Y=2(X²+[4/2]X)-3 =2(X²+2X)-3
Y=2(X+1)²-3-(2)(1)²=2(X+1)²-5
Vertex is at X=-1 Y=-5 or (-1,-5)

So watch what we did with the k term.  
Lets expand 2(X+1)²

2X²+4X+2So we gained a 2 in the expansion of our modified equation so we have to also subtract a 2.
So our form becomes A(X-h)²+k
k is calculated by taking C-A(h)²

So lets do another example:
Y=-3X²-12X+4
Factor the -3 out of the first and second term:
-3(X²+4X)+4
Complete the square:
-3(X+2)²+4-(-3)(2)²=-3(X+2)²+16
The vertex is at:  X=-2  Y=16 or (-2,16)

Practice:
Y=6X²+18X+2
Y=-2X²+5X+3
Y=-7X²-21X+14

Answers posted in next post:

3 Times is a charm

3 divisibility rule

If a number  is divisible by 3, then the sum of the digits of the number are also divisible by 3.

As an example:

63, 123, 2223 are all divisible by 3.  We can easily determine this in our heads without a calculator by adding the digits and then dividing by 3 as follows:

6+3=9 

9÷3=3

63 is evenly divisible by 3.

If we use a calculator, we find 63÷3=21

1+2+3=6

6÷3=2

123 is divisible by 3.

If we use our a calculator, we find 123÷3=41

2+2+2+3=9

9÷3=3

2223 is divisible by 3
If we use our calculator, we find 2223÷3=741

89, 173, and 3113 are not divisible by 3.  Here is why:

8+9=17 and 17 is not divisible by 3

1+7+3=11 and 11 is not divisible by 3

3+1+1+3=8 and 8 is not divisible by 3

This is a very simple rule yet very powerful.  There are other rules that I will present a little later but this is one of the most important rules and this will help you through many tests like the SAT and ACT.

Try these on your own:

Are these numbers divisible by 3?  Add the digits and divide.  Check with your calculator.

1234, 174, 12761, 25437

Hint all are except 12761.

Prime Time 

What are prime numbers?

A prime number is a positive integer that can only be divided evenly by by 2 different integers 1 and itself.  5 can only be divided evenly by 5 and 1 so this number is prime.  6 can be divided evenly by 2, 3, 6, and 1 so 6 would not be a prime number.

Some fun facts:

• 1 is not a prime number.  It is defined as not prime and is not divisible by 2 different integers.  It is only divisible by 1.
• 2 is the only even prime number.
• Numbers that are greater than one and not prime are called composite numbers.
• There are infinitely many prime numbers.
• 0 is not a prime number.

How do we determine if a number is prime?

You can easily figure out if a number is prime by dividing by smaller prime numbers to see if it can be evenly divided.  If I want to find out if 11 prime, I can try to divide it by smaller prime numbers like 2,3,5, or 7.  11 cannot be evenly divided by any smaller prime factors therefore it is prime.  If I wanted to find out if 33 is prime, I can divide 33 by 3 and I find it will evenly divide into 33 with factors of 1,3,11, and 33, therefore it is not prime.

Following are the prime numbers less than 100:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Practice:

Find out if the following numbers are prime:  101, 123 149, and 171.

Definitions:

Divide evenly:  To divide evenly means that if you divide one number by another there is no remainder.  We can make this easier by saying that it will not produce a non integer.  4 divided by 3 = 1.33333..... which is not an integer so 4 is not evenly divisible by 3.