Question

Factorise by grouping method:

$(ax+by{)}^2+(bx-ay{)}^2$

Answer 1

$(ax+by{)}^2+(bx-ay{)}^2$

$=\left[\right(ax{)}^2+(by{)}^2+(2\times ax\times by\left)\right]+\left[\right(bx{)}^2+(ay{)}^2-(2\times bx\times ay\left)\right]$

$\text{Using the identities}$ $(a+b{)}^2=(a^2+b^2+2ab)$,

$\text{and}$ $(a-b{)}^2=(a^2+b^2-2ab\left)\right]$

$=a^2{x}^2+b^2{y}^2+2abxy+b^2{x}^2+a^2{y}^2-2abxy$

$=a^2{x}^2+b^2{x}^2+b^2{y}^2+a^2{y}^2$

$=x^2(a^2+b^2)+y^2(a^2+b^2)$

$=(a^2+b^2)(x^2+y^2)$