Question 39890
let's see...
{{{(sqrt( 2y+7 )) + 4 = y }}}
First we need to isolate the radical...
So we can subtract 4 from both sides of the equation...
{{{(sqrt( 2y+7 )) = y - 4 }}}
Then, in order to eliminate the radical sign, we need to square both sides...
{{{(sqrt( 2y+7 ))^2 = (y - 4)^2 }}}
The square root of a squared number, is just that number so...
{{{ 2y+7 = (y - 4)^2 }}}
Now we can work on the other side by distributing the power of 2...
{{{ 2y+7 = y^2 + 16 }}}
Now we can write the quadratic equation in standard form on one side of the equation and equal to zero by subtracting the 2y and the 7 from both sides...
{{{ y^2 - 2y + 9 = 0 }}}
Now we can solve this equation using the quadratic formula...
{{{y = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 
Now we simply insert the coefficients of our equation such that a=1 b=-2 c=9...
{{{y = (2 +- sqrt( (-2)^2-4*1*9 ))/(2*1) }}}
and solve...
{{{y = (2 +- sqrt( 4 - 36 ))/(2) }}}
and...
{{{y = (2 +- sqrt( - 32 ))/(2) }}}
use imaginary numbers to sove for a negative square root...
{{{y = (2 +- i) sqrt( 32 )/(2) }}}
or...
{{{y = (2 +- i) sqrt( 16*2 )/(2) }}}
the square root of 16 is 4 so that can come out from under the radical...
{{{y = (2 +- 4i) sqrt( 2 )/(2) }}}
Now we can factor out a 2 from the numerator in order to cancel out the 2 in the denominator...
{{{y = 2(1 +- 2i) sqrt( 2 )/(2) }}}
Which can be simplified as...
{{{y = (1 +- 2i) sqrt( 2 )}}}
This is your answer in its simplist form.
So the solution set would be...
{{{(1 + 2i) sqrt( 2 )}}},{{{(1 - 2i) sqrt( 2 )}}}
I hope this helps
Good Luck!