Question

find all real solutions of the quadratic equation. (enter your answers as a comma - separated list. if there is no real solution, enter no real solution.)
x^2 - 14x + 17 = 0
x=

5.
find the domain of the function. (enter your answer using interval notation.)
f(x)=\frac{x^4}{x^2 + x - 72}

Explanation:

Step1: Identify quadratic formula coefficients

For the quadratic equation $x^{2}-14x + 17=0$, where $a = 1$, $b=-14$, $c = 17$.

Step2: Calculate the discriminant $\Delta$

$\Delta=b^{2}-4ac=(-14)^{2}-4\times1\times17=196 - 68=128$.

Step3: Apply the quadratic formula

$x=\frac{-b\pm\sqrt{\Delta}}{2a}=\frac{14\pm\sqrt{128}}{2}=\frac{14\pm8\sqrt{2}}{2}=7\pm4\sqrt{2}$.

Step4: Find the domain of the function

For the function $f(x)=\frac{x^{4}}{x^{2}+x - 72}$, we set the denominator equal to zero: $x^{2}+x - 72=0$.
Factor the quadratic: $(x + 9)(x - 8)=0$. So $x=-9$ or $x = 8$.
The domain is all real - numbers except $x=-9$ and $x = 8$, which in interval notation is $(-\infty,-9)\cup(-9,8)\cup(8,\infty)$.

Answer:

$7 + 4\sqrt{2},7 - 4\sqrt{2}$; $(-\infty,-9)\cup(-9,8)\cup(8,\infty)$