1 Determine the number of solutions of the equation \( y=x^{2}+3 x+5 \)

To determine the number of solutions of the equation \( y = x^2 + 3x + 5 \), we can analyze the discriminant of the quadratic. The standard form of a quadratic equation is \( ax^2 + bx + c \), where \( a = 1 \), \( b = 3 \), and \( c = 5 \). The discriminant \( D \) is given by the formula \( D = b^2 - 4ac \). Plugging in the values: \[ D = 3^2 - 4 \cdot 1 \cdot 5 = 9 - 20 = -11 \] Since the discriminant is negative (\( D < 0 \)), the quadratic equation does not have any real solutions. This means the graph of the equation \( y = x^2 + 3x + 5 \) does not intersect the x-axis, indicating that the parabola opens upwards and lies entirely above the x-axis.