SHORTCUTS TO SQUARE NUMBERS

SHORTCUT 1

TO SQUARE A 2 DIGIT NUMBER THAT ENDS IN 5

EXAMPLE 1

STEP 1:

• Multiply the first digit of the number with a number which is greater than the first number by one. For example:25

• 2 is the first digit number. The number that comes after 2 is 3 on the number line. Now multiply 2 & 3. The answer is 6.
• Now add 25 at the end of the result.
• The answer is 625.

HOW TO FACTORIZE A QUADRATIC TRINOMIAL

EXAMPLE 1

z² + 5z + 6

STEP 1

Multiply the numerical coefficient of the first term with the numerical coefficient of the last term. Here the numerical coefficient of  z²   is 1 since  z²   can also be written as 1 z²   . The numerical coefficient of 6 is 6. In simple words just multiply the numbers  that come before the variables in the first and last terms. Here 6 doesn't have a variable so you obviously need to take 6 as the numerical coefficient

STEP 2

The product of 6 & 1 is 6 that is 6*1=6.

Step 3 

Make a factor tree for the product or list down the factors

6 = 6*1 or 6 = 3*2.

Step 4

Now we have 2 options(6*1 or 3*2) we need to select the one which when added or subtracted gives the result or numeric value of the center term in the equation. Here it is 5 from the term 5z. Let us take the first option 6*1. When you add 6 & 1 you get 7. When you subtract 1 from 6 you get 5. So instead off writing 5z in the main equation we can write it as 6z-1z.

STEP 5

z² +6z-1z+6                

Now factorize

z² +6z

z(z+6)

Now factorize -1z+6

1(-z+6)

Our aim is to get a common term. z+6 & -z+6 aren't same.

STEP 6

So we try out the second option that is 3*2. 3+2 = 5. Therefore instead of 5z we can write it as 3z + 2z

Step 7 

z² +3z+2z+6

Factorize 

z² +3z    

z(z+3)

Factorize

2z+6

2(z+3)

Now we have a common term that is z+3 to factorize.

z(z+3)

2(z+3)

Thus the answer or factorized form of the equation  z² + 5z + 6  is (z+3) (z+2)