Question

$x^2 - 15x+81 > 0$

Answer 1

$(-\infty, \infty)$

Answer 2

The given inequality is $x^2 - 15x + 81 > 0$. To solve this quadratic inequality, we consider the quadratic function $f(x) = x^2 - 15x + 81$. This is a parabola that opens upwards since the coefficient of $x^2$ is $1 > 0$. We find the discriminant of the quadratic equation $x^2 - 15x + 81 = 0$. The discriminant is given by $D = b^2 - 4ac$, where $a=1$, $b=-15$, and $c=81$. $D = (-15)^2 - 4(1)(81) = 225 - 324 = -99$. Since the discriminant $D < 0$, the quadratic equation $x^2 - 15x + 81 = 0$ has no real roots. Since the parabola opens upwards ($a > 0$) and has no real roots, the entire parabola lies above the x-axis. This means that the value of the quadratic expression $x^2 - 15x + 81$ is always positive for all real values of $x$. Therefore, the inequality $x^2 - 15x + 81 > 0$ is true for all real numbers $x$.

The solution set is $(-\infty, \infty)$.