Quadratic EquationHard
Question
If a + b + c = 0, then the quadratic equation 3ax2 + 2bx + c = 0has
Options
A.at least one root in (0,1)
B.one root in (2,3) and the other in (-2,-1)
C.imaginary roots
D.None of the above
Solution
Let f(x) = ax3 + bx2 + cx + d
∴ f (0) = d and f (1) = a + b + c + d = d (∵ a + b + c = 0)
∴ f(0) = f(1)
f is contnuous in the closed interval [0.1] and f is derivable in the open interval (0,1).
Also, f(0) = f(1)
∴ By Roll′s theorem,
f′(a) = 0 for 0 < a < 1
Now, f′(x)= 3ax3 + 2bx + c
⇒ f′(a) = 3aα2 + 2bα + c = 0
∴ Eq. (i) has exist at least one root in the interval (0,1).
Thus, f′(x) must have root in the interval (0,1). or 3ax2 + 2bx + c = 0 has root ∈ (0,1)
∴ f (0) = d and f (1) = a + b + c + d = d (∵ a + b + c = 0)
∴ f(0) = f(1)
f is contnuous in the closed interval [0.1] and f is derivable in the open interval (0,1).
Also, f(0) = f(1)
∴ By Roll′s theorem,
f′(a) = 0 for 0 < a < 1
Now, f′(x)= 3ax3 + 2bx + c
⇒ f′(a) = 3aα2 + 2bα + c = 0
∴ Eq. (i) has exist at least one root in the interval (0,1).
Thus, f′(x) must have root in the interval (0,1). or 3ax2 + 2bx + c = 0 has root ∈ (0,1)
Create a free account to view solution
View Solution FreeMore Quadratic Equation Questions
Difference between the corresponding roots of x2 + ax + b = 0 and x2 + bx + a = 0 is same and a × b, then...Let $M = 3x^{2} - 8xy + 9y^{2} - 4x + 6y + 13$, where $x,y \in R$, then...If the quadratic equations 3x2 + ax +1 = 0 and 2x2 + bx + 1 = 0 have a common root,then the value of the expression 5ab ...If $a,b,c,p,q,r$ are non-zero real numbers, such that $a < b < c$ and $f(x) = (x - a)(x - b)(x - c) - p^{2}(x - a)...If 8, 2 are roots of the equation x2 + ax + β = 0 and 3, 3 are roots of x2 + αx + b = 0 then roots of the equa...