Find all real solutions x^2+3x+3=0

No real solutions.

Step-by-step explanation:

x² + 3x + 3 = 0  This does not factor with integers, so I would use the quadratic formula.

Where ax² + bx + c = 0, x = { -b±√(b²-4ac) } / 2a

   Here a =1, b = 3, and c = 3  

          x = { -3 ± √(3² - 4*1*3) } / 2*1

          x = { -3 ± √(9-12) } / 2

          x = { -3 ± √-3 } / 2

          x = -3/2 + (√-3)/2    OR   x = -3/2 - (√-3)/2

There are no real solutions, because √-3 is not a real number.    

See the "b² - 4ac" part in the Quadratic Formula? Whenever that is negative, there are no real solutions, because you get a square root of a negative. There are no square roots for negative numbers in the Real number system.

I have included a graph in the xy-plane as a picture. See how the graph never touches the x-axis? That is another way for you to know there are no real solutions.

Hope this helps.