Question 1179019
The rational expression (x^2+9x+k)/(x^2+5x-6), x is NOT equal to 1, -6 simplifies to (x+3)/(x-1) because k equals ....
<pre><b>
{{{(x^2+9x+k)/(x^2+5x-6)}}} will simplify to {{{(x+3)/(x-1)}}} if and only if 
every value of x (other than 1 and -6) when substituted in both, gives the
same result.  Therefore, since 0 is the easiest possible number to substitute
for x, let's use it. [But any other number (other than 1 and -6) will work
just as well.  It just won't be as easy.]

Substituting x=0 in

{{{(x^2+9x+k)/(x^2+5x-6)}}} gives

{{{((0)^2+9(0)+k)/((0)^2+5(0)-6)}}}

which reduces to

{{{k/(-6)}}}

Substituting x=0 in {{{(x+3)/(x-1)}}} gives

{{{((0)+3)/((0)-1)}}} 

which reduces to

{{{(3)/(-1)}}} 

or 

{{{-3}}}

Setting those equal:

{{{k/(-6)}}}{{{""=""}}}{{{-3}}}

{{{k}}}{{{""=""}}}{{{18}}}   <--ANSWER

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Check by substituting k=18 in

{{{(x^2+9x+k)/(x^2+5x-6)}}}

{{{(x^2+9x+18)/(x^2+5x-6)}}}

Factoring numerator and denominator:

{{{((x+3)(x+6))/((x+6)(x-1))}}}

Canceling:

{{{((x+3)(cross(x+6)))/((cross(x+6))(x-1))}}}

So it reduces to:

{{{(x+3)/(x-1)}}}

Edwin</pre></b>