If I could borrow your time for a beginner question -- what is the problem with the computation 
?
The problem is looking for the initial velocity of the projectile launched at an elevation of 1 meter with an angle of 45 degrees. The target range is 8 meters and elevated 3 meters.
The initial velocity is supposedly 10.22 m/s. But something went wrong in the middle.
Thank you!
There's only a small mistake: the sign of $g$. Let's look at the quation that you have defined: $$ y=y_0 + v_0 \sin{\theta} t - \frac{1}{2}gt^2 $$ It seems like $y$ is the coordinate that grows upwards, and if $g>0$, this is the equation of a falling object (when $t$ is large, the $y$-coordinate becomes more and more negative). But for some reason, there's also $$ g = -9.8~\frac{\text{m}}{\text{s}^2} $$ In this case, the object would go up indefinitely!
So as a conclusion, just switch the sign of $g$ and it's all good.