Expand And Factorize Quadratic Expressions
Expanding Quadratic Expressions:
Quadratic expressions are algebraic expressions where the variable has an exponent of 2.
For example: x2 + 3x + 4
To expand quadratic equations, use the FOIL (First, Outside, Inside, Last) method.
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
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
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 )
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