`(a+b+c)^2-(ab+bc+ca)`

If 1, omega, omega^(2) are three cube roots of unity, show that (a + omega b+ omega^(2)c) (a + omega^(2)b+ omega c)= a^(2) + b^(2) + c^(2)- ab- bc - ca

If a, b, c are distinct positive real numbers such that the quadratic expression Q_(1)(x) = ax^(2) + bx + c , Q_(2)(x) = bx^(2) + cx + a, Q_(3)(x) = cx^(2) + ax + b are always non-negative, then possible integer in the range of the expression y = (a^(2)+ b^(2) + c^(2))/(ab + bc + ca) is

Resolve into linear factors a^2+b^2+c^2-ab-bc-ca

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

STATEMENT-1 : If a, b, c are distinct and x, y, z are not all zero and ax + by + cz = 0, bx + cy + az = 0, cx + ay + bz = 0 , then a + b + c = 0 and STATEMENT-2 : a^2 + b^2 + c^2 > ab + bc + ca , if a, b, c are distinct.

If a,b,c are non-zero real number such that |(bc,ca,ab),(ca,ab,bc),(ab,bc,ca)|=0, then

If a,b,c are non-zero real number such that |(bc,ca,ab),(ca,ab,bc),(ab,bc,ca)|=0, then

Using column method , add ab+2bc - ca and 2ab - bc -ca and subtract 4ab + 5bc - 3ca

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

If the quadratic equation alpha x^2 +beta x+a^2+b^2+c^2-ab-bc-ca=0 has imaginary roots then