Understanding Engineering Mathematics

Two very important special cases of products of linear expressions are

(a−b)(a+b)=a^2 −b^2

(a+b)^2 =a^2 + 2 ab+b^2

You should know these backwards – literally, given either side you should immediately
be able to write down the other side.

Examples

Fill in the blanks:

(x− 3 )(x+ 3 )= (expanding)

(x+ 2 )^2 = (expanding)

=x^2 − 4 x+ 4 (factorising)

=x^2 − 16 (factorising)

Solution to review question 2.1.1

(i) You just need to be careful with signs here. Take your time until you

are fully confident:

−(x− 3 )=(− 1 )(x− 3 )=(− 1 )x−(− 1 ) 3

=−x+ 3

(ii) 2x(x− 1 )= 2 x×x− 2 x= 2 x^2 − 2 x

(iii) Here you have to remember to collect terms together at the end. We

have

(a− 3 )( 2 a− 4 )=a( 2 a− 4 )− 3 ( 2 a− 4 )

= 2 a^2 − 4 a− 6 a+ 12

= 2 a^2 − 10 a+ 12

(iv)(t− 3 )(t+ 3 )=t^2 + 3 t− 3 t− 32

=t^2 − 9

This is just the difference of two squares result. Practice this until

you can miss out the intermediate step.

(v)(u− 2 )^2 =(u+(− 2 ))^2 =u^2 + 2 (− 2 )u+(− 2 )^2

=u^2 − 4 u+ 4

Again, this square of a linear term is so important that you should

practice it until you are able to jump the intermediate steps and able

to go from ‘right to left’ just as proficiently.