How do you solve (x+5)/(x^2-6x-27)+(x-17)/(x^2-18x+77)=(x+1)/(x^2-14x+33) and check for extraneous solutions?

Feb 13, 2017

See the explanation and the supportive Socratic graphs. T would review my answer, for self-correction, if necessary.

Explanation:

Note that the function is of the form

$\frac{b i \quad r a t i c}{q u \int i c}$

Equating numerator to 0, we get utmost four real zeros.

Equating the quintic to zero, we get utmost five vertical asymptote.

Of course, the x-axis y = 0 is the horizontal asymptote.

The graphs reveal the vertical asymptotes

$\uparrow x = - 3 , 3 , 7 , 9 \mathmr{and} 11 \downarrow$

and the horizontal asymptote

$\leftarrow y = 0 \rightarrow$

By graphical root-bracketing method,, the zeros correct to 2-sd are

x = -3.9, 3.8, 8.6 and 21.4.

Use numerical iterative methods to improve precision to more sd,

with these approximations as starters, for iteration.
.

graph{(x+5)/((x-9)(x+3))+(x-17)/((x-7)(x-11)-(x+1)/((x-3)(x-11)) [-50, 50, -25, 25]}
graph{(x+5)/((x-9)(x+3))+(x-17)/((x-7)(x-11)-(x+1)/((x-3)(x-11)) [-5, 5, -2.5, 2.5]}