Question

Factorise each of the following expressions:
(i) $16x^2+24xy+9y^2$
(ii) $y^2+6xy+9x^2$
(iii) $2x^2-4x+2$
(iv) $20x^2+100x+125$
(v) $a^2+6ab+9b^2-c^2$
(vi) $x^2-y^2+6x+9$
(vii) $m^2-4n^2+10m+25$
(viii) $27a^2+72ab+48b^2$
(ix) $49y^2-70yz+25z^2$
(x) $121x^2-154xy+49y^2$
(xi) $y^2+20y+100$
(xii) $36m^2+60m+25$

Answer 1

$\left(i\right)16x^2+24xy+9y^2$

$=16x^2+12xy+12xy+9y^2$

$=4x(4x+3y)+3y(4x+3y)$

$=(4x+3y)(4x+3y)$

$={(4x+3y)}^2$

$\left(ii\right)y^2+6xy+9x^2$

$=y^2+3xy+3xy+9x^2$

$=y(y+3x)+3x(y+3x)$

$=(y+3x)(y+3x)$

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

$\left(iii\right)2x^2-4x+2$

$=2x^2-2x-2x+2$

$=2x(x-1)-2(x-1)$

$=(x-1)(2x-2)$

$=2(x-1)(x-1)$

$=2{(x-1)}^2$

$\left(iv\right)20x^2+100x+125$

$=5(4x^2+20x+25)$

$=5(4x^2+10x+10x+25)$

$=5(2x(2x+5)+5(2x+5))$

$=5(2x+5)(2x+5)$

$=5{(2x+5)}^2$

$\left(v\right)a^2+6ab+9b^2-c^2$

$=a^2+3ab+3ab+9b^2-c^2$

$=a(a+3b)+3b(a+3b)-c^2$

$=(a+3b)(a+3b)-c^2$

$={(a+3b)}^2-c^2$

$=(a+3b+c)(a+3b-c)$

$\left(vi\right)x^2-y^2+6x+9$

$=x^2+6x+9-y^2$

$=x^2+3x+3x+9-y^2$

$=x(x+3)+3(x+3)-y^2$

$=(x+3)(x+3)-y^2$

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

$=(x+3+y)(x+3-y)$

$=(x+y+3)(x-y+3)$

$\left(vii\right)m^2-4n^2+10m+25$

$=m^2+10m+25-4n^2$

$=m^2+5m+5m+25-4n^2$

$=m(m+5)+5(m+5)-4n^2$

$=(m+5)(m+5)-4n^2$

$={(m+5)}^2-{\left(2n\right)}^2$

$=(m+5+2n)(m+5-2n)$

$=(m+2n+5)(m-2n+5)$

$\left(viii\right)27a^2+72ab+48b^2$

$=3(9a^2+24ab+16b^2)$

$=3(9a^2+12ab+12ab+16b^2)$

$=3(3a(3a+4b)+4b(3a+4b))$

$=3\left((3a+4b)(3a+4b)\right)$

$=3{(3a+4b)}^2$

$\left(ix\right)49y^2-70yz+25z^2$

$=49y^2-35yz-35yz+25z^2$

$=7y(7y-5z)-5z(7y-5z)$

$=(7y-5z)(7y-5z)$

$={(7y-5z)}^2$

$\left(x\right)121x^2-154xy+49y^2$

$=121x^2-77xy-77xy+49y^2$

$=11x(11x-7y)-7y(11x-7y)$

$=(11x-7y)(11x-7y)$

$={(11x-7y)}^2$

$\left(xi\right)y^2+20y+100$

$=y^2+10y+10y+100$

$=y(y+10)+10(y+10)$

$={(y+10)}^2$

$\left(xii\right)36m^2+60m+25$

$=36m^2+30m+30m+25$

$=6m(6m+5)+5(6m+5)$

$=(6m+5)(6m+5)$

$={(6m+5)}^2$