Perfect SquaresPerfect Squares

By: Jenny, Sandra and Temi

Step 1: What is Squaring?

About Squaring

• Squaring is when we multiply a number by itself. Therefore perfect squares are numbers that can be square rooted.

• Some examples would be 49, 10000, 121, 0.36, 1/4, because they can all be square rooted.

About Squaring

• When you apply this to binomials, the results of these equations can be factored into the same binomial. Some examples of a perfect square for binomials would be: (x^2 + 6x + 9), (x^9 + 14x^3 + 49).

Step 2: Calculating Perfect Squares

Calculating Perfect Squares

• There are three simple rules to calculating perfect squares.

1.You square the first term of the binomial2.You multiply both terms of the binomial and

multiply that by two3.You square the last term of the binomial

Calculating Perfect Squares

• There is also a formula that you can use when you are calculating perfect squares.

• When it’s addition the formula is like this:• (a + b)^2 = ( a^2 + 2ab + b^2)• When it’s subtraction, the formula is like this:• (a – b)^2 = (a^2 – 2ab + b^2)

Calculating Perfect Squares

• For example, the question (2x + 5)^2• You substitute “a” with 2x and “b” with 5. • So you’ll get (2x + 5)^2 = (2x)^2 + 2(2x)(5) +

(5)^2• Then all you need to do is to calculate the rest

of the equation. • Show your work and it should be like this:

Calculating Perfect Squares

(2x + 5)^2 Substitute 2x and 5 into the formula:

= (2x)^2 + 2(2x)(5) + (5)^2Calculate: = 4x^2 + 20x +25

Step 3: How do you Factor?

Factoring Back to Perfect Squares

• Of course when you can multiply this: (2x + 5)^2 into this: 4x^2 + 20x +25

• Then you should learn how to factor this: 4x^2 + 20x +25 back into the perfect square it was before:

(2x + 5)^2

Factoring Back to Perfect Squares

• When you are trying to factor a math question, you look at the first term and then the last term.

• If both of them can be square rooted, then you can try to square root the two numbers, multiply the results then double it. If you try that with the above situation then you will see that it works back to this:

Factoring Back to Perfect Squares

Here’s a question for you to factor: 9x^2 – 42x + 49

Now to factor it back: Both terms can be square rootedThe square root for 9x^2 is: 3xThe square root for 49 is: 7

Factoring Back to Perfect Squares

Multiply the two square roots, and then double their product, so it should be like this:

3x x 7 x 2The answer is 42x, which is the second term from

our question. Now you know that it can be factored back into a perfect square. All you need to do is use the formula backwards.

Factoring Back to Perfect Squares

The formula would be like this: For addition: a^2 + 2ab + b^2 = (a + b)^2For subtraction: a^2 – 2ab + b^2= (a – b)^2Since our question is 9x^2 – 42x + 49, so all we need to do now is to substitute everything into the formula.

AND THE ANSWER IS:

(3x – 7)^2

Perfect Squares

• Here is an example of this:• http://www.youtube.com/watch?v

=qPgO1ef1B8s&feature=related

Three questions to ask yourself

1.Is the first term a square?2.Is the third term a square?3.Is the second term twice the product of the

roots?