Assignmax \( x^{2}+3 x+3=0 \)

To solve the quadratic equation \( x^2 + 3x + 3 = 0 \), we can use the quadratic formula, which is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = 3 \), and \( c = 3 \). Plugging in these values, we first calculate the discriminant \( b^2 - 4ac = 3^2 - 4 \times 1 \times 3 = 9 - 12 = -3 \). Since the discriminant is negative, we will have complex solutions! These can be expressed as \( x = \frac{-3 \pm \sqrt{-3}}{2} \), simplifying to \( x = \frac{-3 \pm i\sqrt{3}}{2} \). Just think about it: those complex solutions mean the graph of the equation doesn’t cross the x-axis at all! Instead, it opens up as a U-shape, floating above the x-axis. Isn’t it intriguing how even in math, a little creativity leads to complex numbers? Plus, knowing this lets you understand more about where certain types of equations can lead you. If you’re itching to dig deeper into quadratics or even polynomials in general, exploring the famous "Completions of Squares" method can be a game changer! It's not only a handy technique for these kinds of equations, but it also emerges in various math realms, from calculus to function analysis. And hey, don't shy away from looking into the fascinating history of how mathematicians grappled with and eventually embraced the concept of imaginary numbers!