If the equation `a x^2+b x+c=x` has no real roots, then the equation `a(a x^2+b x+c)^2+b(a x^2+b x+c)+c=x` will have a. four real roots b. no real root c. at least two least roots d. none of these

Options

A. four real roots
B. no real root
C. al least two real roots
D. None of these

Answer

2

Explanation

Since ` ax^(2) + bx + c = x ` has no real roots
`rArr ax^(2) + bx + c gt x or ax^(2) + bx + c lt x ` for all real x .
Let `ax^(2) + bx + c gt x and ax^(2) + bx + c = p(x)`
` therefore P(x) gt x ` for all real x,
`rArr P(P(x)) gt P(x)`
`rArr a(ax^(2) + bx + c )^(2) + b (bx^(2) + bx + c) + c gt ax^(2) + bx + c gt x `
Hence , given equation has no real solution.
Similarly it can be proved for `ax^(2) + bx + c lt x `