a 2 + 2ab + b2 – c2 type expressions

5. a ^{2}\pm2ab+b^{2}-c^{2} type expression

This is advance form of  a^{2}- b^{2} expression. We can say it is combination of 3 & 4 types above. First part involving a and b variables is a whole square. When it is written in whole square form – the whole expression converts to  a^{2}- b^{2} form.

We know that ;

a ^{2}\pm2ab+b^{2}-c^{2}

=(a\pm b) ^{2}-c^{2}

=[(a\pm b) -c][(a\pm b) +c]


Example: Factorize  x^{2}+6x+9 -4 y^{2}


 x^{2}+6x+9 -4 y^{2}

In this expression we can see that first three terms can be taken as whole square of (x+3) by using  a^{2}+2ab+ b^{2}= (a+b)^{2}


 x^{2}+6x+9 -4 y^{2}

=(x+3) ^{2}-4y ^{2}     ;    Now use  a^{2}- b^{2}=(a+b)(a-b)

=(x+3) ^{2}-(2y) ^{2}