Quadratics and Polynomials

Perfect Square

Difference of two squares

Remember Expanding and factorising

What is a perfect square?

A perfect square is a number which can be

expressed as a square:

9 (because 32 = 9)

16 (because 42 = 16)

81 (because 92 = 81)

We could also use expressions!

Perfect Square

Therefore, any expression, such as:

a2

(a+b)2

(k-h)2

Is a perfect square!!!

Why is this helpful?

Try expanding this perfect square:

(c + d)2 = (c + d)(c + d) (now the Distribution Law)

= c2 + cd + dc + d2 (Commutative Law: ab = ba)

= c2 + cd + cd + d2

= c2 + 2cd + d2

What ?!?

That doesn’t look like a quadratic!

OK, expand:

(x+7)2

x2 + 14x +49

So, if you see a quadratic that could be

factorised using Perfect Squares, use it!

x2 + 2x + 1 = …

Perfect Squares are helpful!

Particularly when working with areas:

This is a perfect square:

Area of a square? A = x2

x

x

How to find an area of a square with sides: (a + b)

We use perfect squares in optimisation

problems. How to find a maximum area

covered by the (a+b) square:

(or – how to find an area when you add a slice

of size b to the square a)

ab

ab

a + b

???

???

a + b (side width)

ab

aba2

b2

So, the area of the entire shape is a sum of

individual areas of each shape…

Area of a square with side a+b=

Area of a square = its side squared (a+b)2

ab

ab

a2

b2

a + b (side width)

Area:

(a+b)2 = a2 + 2ab + b2

(a+b)2 = a2 + 2ab + b2

This is the general model for expanding a

perfect square. It means that we can expand

them very easily.

e.g. (x + 3)2 = x2 + 2*3*x + 32

= x2 + 6x + 9

Try these

Try these

1) (x + 2)2 =

2) (x + 4)2 =

3) (x + 1)2 =

Application Problem

• A quadrangle has one side four units longer

than the other. Its area is 60 square units.

What are the dimensions of the quadrangle?

If we denote the length of one side of the

quadrangle as x units, then the other

must be x + 4 units in length.

We must solve the equation: x *( x+4) =

60, which is equivalent to solving the

quadratic equation:

x2 + 4x -60 = 0

What about negative?

What about when there is a negative square?

e.g. (x - 3)2

Can our area model still work? How would

we label it?

(x-3)2

What does it mean: reduce square x by a slice

of size 3.x

a

Difference of two squares

• One Square: x2

• Second Square: 42

• Difference: subtraction

• Difference of two squares:

x2 – 4

Factorise?

Factorise

x2 – 4 = 0

x2 = 4

x1,2 = √4

x1,2 = ± 2

x1 = 2, x2 = -2

Hence:

x2 – 4 = (x-2)(x+2)