Quadratic EquationHard
Question
If the equation $ax^{2} + bx + c = x,a,b,c \in R$ and $a \neq 0$, has no real roots, then the equation $a\left( ax^{2} + bx + c \right)^{2} + b\left( ax^{2} + bx + c \right) + c = x$ will have
Options
A.2 distinct real roots
B.no real roots
C.2 equal real roots
D.nothing can be said
Solution
Let $P(x) = {ax}^{2} + bx + c$
$P(x) = x$ has no real roots
$$\begin{matrix} & \ \Rightarrow \ P(x) > x\forall x \in R \\ & \ \Rightarrow \ P(P(x)) > P(x) > x\forall x \in R \end{matrix}\ \text{~or~}\ \begin{matrix} & P(x) < x \\ & P(P(x)) < P(x) < x\forall x \in R \end{matrix}\ \forall x \in R$$
Hence, $P(P(x)) = x$ has no real roots
Create a free account to view solution
View Solution FreeMore Quadratic Equation Questions
If α and β are the roots of x2 + px + q = 0 and α4, β4 are the root of x2 - rx + s = 0,then the equa...If one root of the equation x2 − x − k = 0 is square of the other, then k equals to -...Let $p(x) = x^{6} + ax^{5} + bx^{4} + x^{3} + bx^{2} + ax + 1$. Given that 1 is a root of $p(x) = 0$ and -1 is not. What...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)...The number of values of k, for which the system of equations :(k + 1)x + 8y = 4kkx + (k + 3)y = 3k - 1has no solution, i...