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
Let the inequalities $y( - 1) > - 4,y(1) < 0$ and $y(3) > 5$ are known to hold for $y = {ax}^{2} + bx + c$, the...If all the roots of the equation $x^{4} - 12x^{3} + ax^{2} + bx + 81 = 0$ where $a,b \in R$ are positive, then...The possible set of real values of ' $a$ ' for which, for all $x$ not exceeding unity in absolute value, the inequality ...Let $a,b$ are two different positive integers and the two quadratic equations $(a - 1)x^{2} - \left( a^{2} + 2 \right)x ...If roots of the equation x2 - bx + c = 0 be two consectutive integers, then b2 - 4c equals...