we have the functions
[tex]\begin{gathered} g(x)=-2x^2+13x+7 \\ h(x)=-x^2+4x+21 \end{gathered}[/tex]
Part A
Equate both equations
[tex]-2x^2+13x+7=-x^2+4x+21[/tex]
Solve for x
[tex]\begin{gathered} -2x^2+13x+7+x^2-4x-21=0 \\ -x^2+9x-14=0 \end{gathered}[/tex]
Solve the quadratic equation
using the formula
a=-1
b=9
c=-14
substitute
[tex]x=\frac{-9\pm\sqrt{9^2-4(-1)(-14)}}{2(-1)}[/tex][tex]x=\frac{-9\pm5}{-2}[/tex]
The values of are
x=2 and x=7
The answer Part A
The distances are x=2 units and x=7 units
Part B
f(x)=g(x)/h(x)
so
[tex]f(x)=\frac{-2x^2+13x+7}{-x^2+4x+21}[/tex]
Rewrite in factored form
[tex]\begin{gathered} f(x)=\frac{-2(x+\frac{1}{2})(x-7)}{-(x+3)(x-7)} \\ \\ f(x)=\frac{(2x+1)}{(x+3)} \end{gathered}[/tex]
The given function has a discontinuity at x=7 (hole), a vertical asymptote at x=-3
and horizontal asymptote at y=2
