Question

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?

Answer 1

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

`(biquadratic)/(quintic)`

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

`uarrx = -3, 3, 7, 9 and 11 darr`

and the horizontal asymptote

`larr y = 0 rarr`

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]}