Question 61003
Find all real solutions to 

(a) {{{x^2+x+sqrt(x^2+x)-2=0}}} 

Use the substitution method.
{{{(sqrt(x^2+x))^2+sqrt(x^2+x)-2=0}}}
Let {{{P=sqrt(x^2+x)}}}
{{{P^2+P-2=0}}}  Factor
(P+2)(P-1)=0
P+2=0 and P-1=0
P=-2 and P=1
But we want to know what x is, so let {{{sqrt(x^2+x)=P}}}
{{{sqrt(x^2+x)=-2}}} and {{{sqrt(x^2+x)=1}}}
{{{(sqrt(x^2+x))^2=(-2)^2}}} and {{{(sqrt(x^2+x))^2=1^2}}}
{{{x^2+x=4}}} and {{{x^2+x=1}}}
{{{x^2+x-4=0}}} and {{{x^2+x-1=0}}} 
You have to use the quadratic equation to solve these:
{{{x=(-b+-sqrt(b^2-4ac))/(2a)}}}
when a=1, b=1, c=-4
{{{x=(-1+-sqrt((1)^2-4(1)(-4)))/(2*1)}}}
{{{x=(-1+-sqrt(1+16))/2}}}
{{{x=(-1+-sqrt(17))/2}}}  You have to check these to see if they are valid or extraneous.  I checked these by plugging them into my calculator and the were both extraneous.
when a=1, b=1, and c=-1
{{{x=(-1+-sqrt((1)^2-4(1)(-1)))/(2*1)}}}
{{{highlight(x=(-1+-sqrt(5))/2)}}}  You have to check these to see if they are valid or extraneous.  I checked these with my calculator and they were both valid.
If you have a TI-83 or 84, I can give you instructions on checking for extraneous solutions if you ask.
Happy Calculating!!!