What are perfect squares and how to factor them?

 

What are they?

• A perfect square trinomial always starts and ends with a square, and looks like (a+b)^2=a^2+2ab+b^2.

• In the expression ax^2+bx+c, the "a" and the "c" are squares, so a number is multiplied by itself to get "a" and "c", however they are whole numbers.

• For example 4x^2+12x+9. 

• 4 is a square (2x2=4) and 9 is a square (3x3=9).

• However, in order for it to be a perect square trinomial, the square roots of "a" and "c" need to equal to "b" when multiplied by 2.

• So, square root of 4 (2), and square root of 9 (3), need to equal to "b" when multiplied by 2.

• 2(2)(3)=b

• 2(6)=b

• 12=b

 

 

Let's factor them!

For example, 25x^2+20x+4

• Is this a perfect square?

• The square root of 25 is 5, and square root of 4 is 2.

• 5 multiplied by 2 is 10.

• 2 multplied by 10 is 20.

• The "b" value is 20, so it is a perfect square.

• We write this as (5x+2) (5x+2).

• So (5x+2)^2.

• We basically write the square roots of both "a" and "c"!

• If we expand this, we must get 25x^2+20x+4.

 

Why do we multiply 2(a)(c)?

We multiply it by 2 because it is a sqaure.

If we expand (5x+2)^2.

• (5x+2) (5x+2)

• 5x multiplied by 5x is 25x^2.

• 5x multiplied by 2 is 10x.

• 2 multiplied by 5x is 10x.

• 2 multiplied by 2 is 4.

• Notice that we have two 10x.

• Therefore, that 2 comes because we have two of the same like terms!

 

 

Practice these....

1) u^2+18u+81

2) f^2+14f+49

3) 64j^2-112jk+49k^2

 

 

Answers are...

 

 

 

 

 

1) (u+9)^2

2) (f+7)^2

3) (8j-7)^2