2.3.3 Quadratic expressions
An expression of the form \( ax^2+bx+ c\) , where \( a≠0 \) is called the quadratic expression.
If \( a=1 \), then it is called a monic quadratic expression
\( x^2+bx+c \)
If \( a≠1 \), then it is called a non-monic quadratic expression
\( ax^2+bx+c \)
Monic Quadratic expression
To factorize a monic quadratic expression \( (x^2+bx+c) \)
We need a pair of numbers that
Multiply to \( c \)
Add to \( b \)
Split the middle term
Take \( x \) out of the first two terms and a common factor from last two terms
Now the expression has a common bracket
Take out the common bracket
Worked example:
Factorise completely.
\( x^2-5x+6 \)
We need two numbers that multiply to \( 6 \) and add to \( -5 \)
\( -3 \) and \( -2 \) satisfy this
Split the middle term
\( x^2-3x-2x+6 \)
Take \( x \) out of the first two terms and \( -3 \) from last two terms
\( x(x-3)-2(x-3) \)
\( (x-3) \) is a common bracket, Take it out
\( (x-3)(x-2) \)
\( (x-3)(x-2) \)
Non-monic Quadratic expression
To factorize a non-monic quadratic expression \( (ax^2+bx+c) \)
We need a pair of numbers that
Multiply to \( ac \)
Add to \( b\)
Split the middle term
Find out the highest common factor of the first two terms and take it out
Find out the highest common factor of the last to terms and take it out
Take out the common bracket
Worked example:
Factorise completely.
\( 6x^2-7x-3 \)
We need two numbers that multiply to \( 6 \times (-3)=-18 \) and add to \( -7 \)
\( -9\) and \( 2 \) satisfy this
Split the middle term
\( 6x^2-9x+2x-3 \)
The highest common factor of the first two terms is \( 3x \) and the highest common factor of the last two terms is \( 1 \)
\( 3x(2x-3)+1(2x-3) \)
\( (2x-3) \) is a common bracket, take it out
\( (2x-3)(3x+1) \)
\( (2x-3)(3x+1) \)