`a, b, c in R, a!= 0` and the quadratic equation`ax^2+bx+c=0` has no real roots, then

Statement -1 If the equation (4p-3)x^(2)+(4q-3)x+r=0 is satisfied by x=a,x=b nad x=c (where a,b,c are distinct) then p=q=3/4 and r=0 Statement -2 If the quadratic equation ax^(2)+bx+c=0 has three distinct roots, then a, b and c are must be zero.

The values of p for which the quadratic equation px^2+2x+p=0 has real roots are:

If a,b, c epsilon R(a!=0) and a+2b+4c=0 then equatio ax^(2)+bx+c=0 has

If 2a+3b+6c = 0, then show that the equation a x^2 + bx + c = 0 has atleast one real root between 0 to 1.

If alpha, beta are the roots of the quadratic equation x^2 + bx - c = 0 , the equation whose roots are b and c , is a. x^(2)+alpha x- beta=0 b. x^(2)-[(alpha +beta)+alpha beta]x-alpha beta( alpha+beta)=0 c. x^(2)+[(alpha + beta)+alpha beta]x+alpha beta(alpha + beta)=0 d. x^(2)+[(alpha +beta)+alpha beta)]x -alpha beta(alpha +beta)=0

Let a,b,c,d be distinct real numbers and a and b are the roots of the quadratic equation x^2-2cx-5d=0 . If c and d are the roots of the quadratic equation x^2-2ax-5b=0 then find the numerical value of a+b+c+d .

If 0 lt a lt b lt c and the roots alpha,beta of the equation ax^2 + bx + c = 0 are non-real complex numbers, then

If ax^(2)+bx+c=0, a,b, c in R , then find the condition that this equation would have atleast one root in (0, 1).

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.

If 2 - i is a root of the equation ax^2+ 12x + b=0 (where a and b are real), then the value of ab is equal to :