Khrrabrov A.

Are there such integers $a$ and $b$ that

a) the equation $x^{2} + a x + b = 0$ has no roots, while the equation $\left[x^{2}\right] + a x + b = 0$ has roots?

b) the equation $x^{2} + 2 a x + b = 0$ has no roots, while the equation $\left[x^{2}\right] + 2 a x + b = 0$ has roots?

#
a) The equation $x^{2}-3 x+3=0$ has no roots, since its discriminant is negative, while the equation [ $\left.x^{2}\right]$ $-3 x+3=0$ has roots $x_{1}=4 / 3$,

$x_{2}=5 / 3$

b) Let the function $f(x)=x^{2}+2 a x=b=(x+a)^{2}+b-a^{2}$ not turn into zero. Then $b-a^{2}$ is positive, and therefore not less than 1 ( $a$ and $b$ - integers). Thus, $f(x) \geq 1$ for any $x$. Replacing $x^{2}$ with its integer part reduces the value of the function by less than 1, and therefore keeps it positive.

## Answer

a) They exist; b) they do not exist.