Explanation:
There is a theorem :
[tex]\mathbf{a^{2}-b^{2}=(a-b)(a+b)}[/tex]
Proof: open the brackets on right side of equality by using distributive property
RHS = a(a+b) - b(a+b)
RHS = [tex]\mathrm{a^{2}+ab-ab-b^{2}}[/tex]
RHS = [tex]\mathrm{a^{2}-b^{2}}[/tex] = LHS
Using the above theorem you can solve every question you asked:
- [tex]\mathrm{4x^{2}-1=(2x)^{2}-1^{2}=(2x-1)(2x+1)}[/tex]
- [tex]\mathrm{9x^{2}-64=(3x)^{2}-(8)^{2}=(3x-8)(3x+8)}[/tex]
- [tex]\mathrm{49x^{2}-121=(7x)^{2}-(11)^{2}=(7x-11)(7x+11)}[/tex]
- [tex]\mathrm{4x^{2}-y^{2}=(2x)^{2}-y^{2}=(2x-y)(2x+y)}[/tex]
- [tex]\mathrm{9x^{2}-b^{2}=(3x)^{2}-b^{2}=(3x-b)(3x+b)}[/tex]
- [tex]\mathrm{16x^{2}-9b^{2}=(4x)^{2}-(3b)^{2}=(4x-3b)(4x+3b)}[/tex]
- [tex]\mathrm{x^{2}-42^{2}=(x-42)(x+42)}[/tex]
