If two roots of `x^3-a x^2+b x-c=0` are equal inn magnitude but opposite in signs, then prove that `a b=c

In equation x^4-2x^3+4x^2+6x-21=0 if two of its roots are equal in magnitude but opposite in sign find the roots.

If the roots of the equation (1)/(x+p)+(1)/(x+q)=(1)/(r ),(x ne -p, x ne -q, r ne 0) are equal in magnitude but opposite in sign, then p+q is equal to

If the roots of ht cubic, x^3+a x^2+b x+c=0 are three consecutive positive integers, then the value of (a^2//b+1) is equal to __________.

If the roots of the cubic, x^2+a x^2+b x+c=0 are three consecutive positive integers, then the value of (a^2//b+1) is equal to __________.

If the roots of (a-b)x^(2)+(b-c)x+(c-a)=0 are real and equal, then prove that b, a, c are in arithmetic progression.

If the roots of (a-b)x^(2)+(b-c)x+(c-a)=0 are equal, prove that 2a=b+c .

If the roots of the quadratic equation 3x^(2)+2x+a^(2)-a=0 in x are of opposite signs, then a lies in the interval

If x is real and the roots of the equation a x^2+b x+c=0 are imaginary, then prove tat a^2x^2+a b x+a c is always positive.

If the sum of the roots of the quadratic equation a x^2+b x+c=0 is equl to the sum of the squares of their reciprocals, then prove that a/c , b/a a n d c/b are in H.P.