Expand And Factorize Quadratic Expressions

Expanding Quadratic Expressions:

Quadratic expressions are algebraic expressions where the variable has a power of 2.

For example: x2 + 3x + 4

To expand quadratic equations, use the FOIL (First, Outside, Inside, Last) method.

First Outside Inside Last

Example One

Expand ( x + 3 ) ( x + 2 ) without and with using FOIL.

Answer (without using FOIL):
( x + 3 ) ( x + 2 )
= x ( x + 2 ) + 3 ( x + 2 )
= x2 + 2x + 3x + 6
= x2 + 5x + 6

Answer (with using FOIL):
( x + 3 ) ( x + 2 )
= x2 + 2x + 3x + 6
= x2 + 5x + 6


Example Two

( x + 4 ) ( x – 2 )
= x2 – 2x + 4x – 8
= x2 + 2x – 8


Example Three

( 2x + 5 ) ( 3x – 8 )
= 6x2 – 16x + 15x – 40
= 2x2 – x – 40

Questions - Expand Using FOIL

Q1. ( x + 6 ) ( x + 5 )
Q2. ( x – 5 ) ( x – 4 )
Q3. ( 2x + 5 ) ( 6x – 2 )

Answers
A1. x2 + 11x + 30
A2. x2 – 9x + 20
A3. 12x2 + 26x – 10

Perfect Squares

( x + a )2 = x2 + 2ax + a2
( x – a )2 = x2 – 2ax + a2

Example Four

( x + 5 )2
= ( x + 5 ) ( x + 5 )
= x2 + 10x + 25


Example Five

( x – 3 )2
= ( x – 3 ) ( x – 3 )
= x2 – 6x + 9

Questions - Expand These Perfect Squares

Q1. ( x + 7 )2
Q2. ( 2x + 5 )2

Answers
A1. x2 + 14x + 49
A2. 4x2 + 20x + 25

Difference of Squares

( x + a ) ( x – a ) = x2 – a2

Example Six

( x + 5 ) ( x – 5 )
= x2 – 5x + 5x – 25
= x2 – 25


Example Seven

( x – 3 ) ( x + 3 )
= x2 – 3x + 3x – 9
= x2 – 9

Questions - Expand These Difference of Squares

Q1. ( x + 7 ) ( x – 7 )
Q2. ( 2x + 5 ) ( 2x – 5 )

Answers
A1. x2 – 49
A2. 4x2 – 25

Factorizing Quadratic Expressions:

Factorizing is the reverse of expanding.



Example Eight

x2 + 6x + 5
= ( x + 5 ) ( x + 1 )


Example Nine

6x2 + 2x – 20
= ( 2x + 4 ) ( 3x – 5 )

Questions - Factorize

Q1. x2 – 7x – 8
Q2. x2 + x – 12

Answers
A1. ( x – 8 ) ( x + 1 )
A2. ( x + 4 ) ( x – 3 )