Question 211413
 I am stumped...I can not figure these two problems out for the life of me...can someone please assist me with these two problems? 
Find all real solutions to each equation. 
<pre><font size = 4 color = "indigo"><b>
{{{x^2 + x + sqrt (x^2 + x) - 2 = 0}}} 

We group the first two terms as one term:

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

Now there are three terms on the left.  Notice 
that if we square the variable of the
middle term {{{sqrt(x^2+x)}}}, we get the 
variable of the first term {{{(x^2+x)}}}.

Therefore we substitute the letter "{{{u}}}" for
the variable of the middle term, that is

{{{sqrt(x^2+x)=u}}}

Then squaring both sides we have

{{{(x^2+x)=u^2}}}

So we substitute and

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

becomes simply

{{{u^2+u-2=0}}}

Which we factor 

{{{(u+2)(u-1)=0}}}

and solve getting {{{u=-2}}} and {{{u=1}}}

Since {{{sqrt(x^2+x)=u}}} and a square root
can never be negative in real numbers, we can 
rule out {{{u=-2}}}. So {{{u=1}}}

Substituting back

{{{(x^2+x)=u^2}}}

becomes

{{{(x^2+x)=1^2}}}
{{{x^2+x=1}}}
{{{x^2+x-1=0}}}

We use the quadratic formula:

 {{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 

 {{{x = (-(1) +- sqrt( (1)^2-4*(1)*(-1) ))/(2*(1)) }}}

 {{{x = (-1 +- sqrt( 1+4 ))/2 }}}

 {{{x = (-1 +- sqrt(5))/2 }}}

--------------------------------

Find all real and imaginary solutions to each equation. 
70) {{{b^4 + 13b^2 + 36 = 0}}}

Notice that if we square the variable of the
middle term {{{sqrt(b^2)}}}, we get the 
variable of the first term {{{(b^4)}}}.

Therefore we substitute the letter "{{{u}}}" for
the variable of the middle term, that is

{{{b^2=u}}}

and squaring both sides:

{{{b^4=u^2}}}

So we substitute and

{{{b^4 + 13b^2 + 36 = 0}}}

becomes simply

{{{u^2+13u+36=0}}}

Which we factor 

{{{(u+9)(u+4)=0}}}

and solve getting {{{u=-9}}} and {{{u=-4}}}

Now since {{{b^2=u}}}, we have

{{{b^2=-9}}} and {{{b^2=-4}}}

Taking square roots:

{{{b=" "+-sqrt(-9)}}} and {{{b=" "+-sqrt(-4)}}}

or

{{{b}}}=±{{{3i}}} and {{{b}}}=±{{{2i}}}

Edwin</pre>