Question

Expand each of the following, using suitable identities:
(i) $(x+2y+4z{)}^2$
(ii) $(2x-y+z{)}^2$
(iii) $(-2x+3y+2z{)}^2$
(iv) $(3a-7b-c{)}^2$
(v) $(-2x+5y-3z{)}^2$
(vi) ${\left[\frac{1}{4}a-\frac{1}{2}b+1\right]}^2$

Answer 1

We know,

$(a+b+c{)}^2=a^2+b^2+c^2+2ab+2bc+2ac$

i) $(x+2y+4z{)}^2$

$=x^2+(2y{)}^2+(4z{)}^2+2\left(x\right)\left(2y\right)+2\left(2y\right)\left(4z\right)+2\left(x\right)\left(4z\right)$

$=x^2+4y^2+16z^2+4xy+16yz+8xz$

ii) $(2x-y+z{)}^2$

$=(2x{)}^2+(-y{)}^2+(z{)}^2+2(2x\left)\right(-y)+2(-y\left)\right(z)+2(2x\left)\right(z)$

$=4x^2+y^2+z^2-4xy-2yz+4xz$

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

$=(-2x{)}^2+(3y{)}^2+(2z{)}^2+2(-2x\left)\right(3y)+2(3y\left)\right(2z)+2(-2x\left)\right(2z)$

$=4x^2+9y^2+4z^2-12xy+12yz-8xz$

iv) $(3a-7b-c{)}^2$

$=(3a{)}^2+(-7b{)}^2+(-c{)}^2+2(3a\left)\right(-7b)+2(-7b\left)\right(-c)+2(3a\left)\right(-c)$

$=9a^2+49b^2+c^2-42ab+14bc-6ac$

v) $(-2x+5y-3z{)}^2$

$=(-2x{)}^2+(5y{)}^2+(-3z{)}^2+2(-2x\left)\right(5y)+2(5y\left)\right(-3z)+2(-3z\left)\right(-2x)$

$=4x^2+25y^2+9z^2-20xy-30yz+12xz$

iv) ${(\frac{1}{4}a-\frac{1}{2}b+1)}^2$

$={\left(\frac{1}{4}a\right)}^2+{(-\frac{1}{2}b)}^2+(1{)}^2+2(\frac{1}{4}a\times -\frac{1}{2}b)+2(-\frac{1}{2}b\times 1)+2(\frac{1}{4}a\times 1)$

$=\frac{1}{16}{a}^2+\frac{1}{4}{b}^2+1-\frac{1}{4}ab-b+\frac{1}{2}a$