Let `a, b, c` be real numbers, `a != 0.` If `alpha` is a zero of `a^2 x^2+bx+c=0, beta` is the zero of `a^2x^2-bx-c=0 and 0 < alpha < beta` then prove that the equation `a^2x^2+2bx+2c=0` has a root `gamma` that always satisfies `alpha < gamma < beta.`

Let a ,b ,c be real numbers with a!=0 and alpha,beta be the roots of the equation a x^2+b x+c=0. Express the roots of a^3x^2+a b c x+c^3=0 in terms of alpha,betadot

If alpha. beta are the roots of x^2 +bx+c=0 and alpha + h, beta + h are the roots of x^2 + qx +r=0 then 2h=

If alpha, beta are the roots of x^(2) + bx-c = 0 , then the equation whose roots are b and c is

Let a,b,c be real numbers in G.P. such that a and c are positive , then the roots of the equation ax^(2) +bx+c=0

Find the values of alpha^(3)+beta^(3) in terms of a, b, c if alpha, beta are roots of ax^(2)+bx+c=0 , c != 0

Let a ,b ,and c be real numbers such that 4a+2b+c=0 and a b > 0. Then the equation a x^2+b x+c = 0

Let a, b, c, p, q be the real numbers. Suppose alpha,beta are the roots of the equation x^2+2px+ q=0 . and alpha,1/beta are the roots of the equation ax^2+2 bx+ c=0 , where beta !in {-1,0,1} . Statement I (p^2-q) (b^2-ac)>=0 Statement 11 b !in pa or c !in qa .

If alpha, beta are the roots of x^(2) + bx + c = 0 and, alpha +h, beta+h are the roots of x^(2) + qx +r = 0 then h =

Form the values of (1)/(alpha)+(1)/(beta) in terms of a, b, c if alpha, beta are roots of ax^(2)+bx+c=0 , c != 0

Let a, b and c be real numbers such that 4a + 2b + c = 0 and ab gt 0. Then the equation ax^(2) + bx + c = 0 has