# 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?

##### 1 Answer

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

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

and the horizontal asymptote

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]}