Multiple choice

If the equation $ax^2+bx+c=x$ has no real roots, then the equation $a(ax^2+bx+c)^2+b(ax^2+bx+c)+c=x$ will have

  1. four real roots

  2. no real roots

  3. at least two real roots

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let f(x) = ax^2 + bx + c. The equation f(x) = x has no real roots, so f(x) - x is never zero. The second equation is f(f(x)) = x. If f(x) = x has no real roots, then f(f(x)) = x implies f(x) must equal x for some x, which is impossible as f(x) - x is never zero.

AI explanation

Let f(x) = ax^2 + bx + c, so the first equation is f(x) = x and the second equation is a[f(x)]^2 + b[f(x)] + c = x, which is exactly f(f(x)) = x. Since f(x) = x has no real roots, f(x) - x is never zero for any real number x. If we subtract x from both sides of the second equation, we get f(f(x)) - x = 0. Notice that f(f(x)) - x can be factored as (f(x) - x) multiplied by another polynomial. Since the factor f(x) - x is never zero, the second factor must provide any real roots, but a detailed analysis shows that if f(x) = x has no real roots, then f(f(x)) = x also cannot have any real roots. The graph of f(f(x)) never intersects the line y = x, so the equation has no real roots.

Keep practicing — related questions