Answer:
(a)
[tex]x^2+64=(x-8i)(x+8i)[/tex]
(b)
[tex]16x^2+49=(4x-7i)(4x+7i)[/tex]
(c)
[tex](x+9i)^2=x^2+18xi-81[/tex]
(d)
[tex](x-2i)^2=x^2-4xi-4[/tex]
(e)
[tex](x+(3+5i))^2=10ix+30i+25i^2+x^2+6x+9[/tex]
Step-by-step explanation:
(a)
we are given
[tex]x^2+64[/tex]
we can also write as
[tex](x)^2+8^2[/tex]
[tex](x)^2-(8i)^2[/tex]
now, we can use factor formula
[tex]a^2-b^2=(a-b)(a+b)[/tex]
we get
[tex]x^2+64=(x-8i)(x+8i)[/tex]
(b)
we are given
[tex]16x^2+49[/tex]
we can also write as
[tex](4x)^2+7^2[/tex]
[tex](4x)^2-(7i)^2[/tex]
now, we can use factor formula
[tex]a^2-b^2=(a-b)(a+b)[/tex]
we get
[tex]16x^2+49=(4x-7i)(4x+7i)[/tex]
(c)
[tex](x+9i)^2[/tex]
we can use formula
[tex](a+b)^2=(a^2+2ab+b^2)[/tex]
so, we can write as
[tex](x+9i)^2=x^2+2x\cdot \:9i+\left(9i\right)^2[/tex]
we can simplify it
and we get
[tex]=x^2+18ix-81[/tex]
[tex]=x^2+18xi-81[/tex]
(d)
[tex](x-2i)^2[/tex]
we can use formula
[tex](a+b)^2=(a^2+2ab+b^2)[/tex]
so, we can write as
[tex](x-2i)^2=x^2-2x\cdot \:2i+\left(2i\right)^2[/tex]
we can simplify it
and we get
[tex]=x^2-4ix-4[/tex]
[tex]=x^2-4xi-4[/tex]
(e)
[tex](x+(3+5i))^2[/tex]
we can distribute
so, we can write as
[tex]=\left(5i+x+3\right)\left(5i+x+3\right)[/tex]
[tex]=xx+x\cdot \:3+x\cdot \:5i+3x+3\cdot \:3+3\cdot \:5i+5ix+5i\cdot \:3+5i\cdot \:5i[/tex]
[tex]=xx+3x+5ix+3x+3\cdot \:3+3\cdot \:5i+5ix+5\cdot \:3i+5\cdot \:5ii[/tex]
now, we can simplify it
[tex]=10ix+30i+25i^2+x^2+6x+9[/tex]
