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.

Statement-1: If a, b, c in R and 2a + 3b + 6c = 0 , then the equation ax^(2) + bx + c = 0 has at least one real root in (0, 1). Statement-2: If f(x) is a polynomial which assumes both positive and negative values, then it has at least one real root.

If 27a+9b+3c+d=0 then the equation 4ax^(3)+3bx^(2)+2cx+d has at leat one real root lying between

If the equation ax^(2) + bx + c = 0, a,b, c, in R have non -real roots, then

If 4a+2b+c=0 , then the equation 3ax^(2)+2bx+c=0 has at least one real lying in the interval

Let a , b , c be nonzero real numbers such that int_0^1(1+cos^8x)(a x^2+b x+c)dx =int_0^2(1+cos^8x)(a x^2+b x+c)dx=0 Then show that the equation a x^2+b x+c=0 will have one root between 0 and 1 and other root between 1 and 2.

Let a , b , c be nonzero real numbers such that int_0^1(1+cos^8x)(a x^2+b x+c)dx =int_0^2(1+cos^8x)(a x^2+b x+c)dx=0 Then show that the equation a x^2+b x+c=0 will have one root between 0 and 1 and other root between 1 and 2.

If a, b, c ∈ R, a ≠ 0 and the quadratic equation ax^2 + bx + c = 0 has no real root, then show that (a + b + c) c > 0

If a, b, c, d are real numbers such that (a+2c)/(b+3d)+(4)/(3)=0 . Prove that the equation ax^(3)+bx^(2)+cx+d=0 has atleast one real root in (0, 1).

If the equation a x^2+b x+c=0 has two positive and real roots, then prove that the equation a x^2+(b+6a)x+(c+3b)=0 has at least one positive real root.

If a, b, c in R and the quadratic equation x^2 + (a + b) x + c = 0 has no real roots then