Prove that: `(a-b)^2=a^2-2ab+b^2`

Prove that |axxb^2 =a^2b^2 - (a.b)^2

Prove that: |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-a^2,ca-b^2]| is divisible by a+b+c and find the quotient.

Prove that : |{:(b^(2)c^(2),bc, b+c),(c^(2)a^(2),ca, c+a),(a^(2)b^(2),ab, a+b):}|=0

Prove that |[(b+c)^2, a^2, bc],[(c+a)^2, b^2, ca],[(a+b)^2, c^2, ab]|=(a-b)(b-c)(c-a)(a+b+c)(a^2+b^2+c^2)

Prove that a^(2)+b^(2)+c^(2)-ab-bc-ca is always non-negative for all values of a,b and c.

Prove that a^(2)+b^(2)+c^(2)-ab-bc-ca is always non-negative for all values of a,b and c

If a,b,c,d are in G.P.prove that: (ab-cd)/(b^(2)-c^(2))=(a+c)/(b)

Prove that det[[a^(2),2ab,b^(2)b^(2),a^(2),2ab2ab,b^(2),a^(2)]]=(a^(3)+b^(3))^(2)

If a,b,c are in G.P.then prove that (a^(2)+ab+b^(2))/(bc+ca+ab)=(b+a)/(c+b)

If a,b,c are in G.P., prove that a^(2)+b^(2),ab+bc,b^(2)+c^(2) are also in G.P