What are the solutions?

1/2x^2+2x+3=0

multiply all sides by 2x^2

1 + 4x^3 + 6x^2 = 0
let y = x^2
1 + 4y^2 + 6y=0
4y^2 + 6y + 1 = 0
Apply the quadratic formula then sqrt the answer to get the answer

the way you typed it,

multiply each term by 2

x^2 + 4x + 6 = 0
complete the square ...
x^2 + 4x + 4 = -6+4
(x+2)^2 = -2
x+2 = ± √-2
x = -2 ± i√2

To find the solutions of the equation 1/2x^2 + 2x + 3 = 0, you can use the quadratic formula. The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

In this equation, a, b, and c are the coefficients of the quadratic equation.

Compared to the standard quadratic equation of the form ax^2 + bx + c = 0, the given equation can be rewritten as:

(1/2)x^2 + 2x + 3 = 0

Here, a = 1/2, b = 2, and c = 3.

Now, substitute these values into the quadratic formula:

x = (-2 ± √(2^2 - 4(1/2)(3))) / 2(1/2)

Simplifying further, we have:

x = (-2 ± √(4 - 6)) / 1

x = (-2 ± √(-2)) / 1

Since the expression under the square root (√) is negative, it means that the quadratic equation has no real solutions. The solutions are imaginary numbers. Specifically, the solutions can be written as:

x = (-2 ± √(-2)i) / 1

where i is the imaginary unit, defined as √(-1).