To find the solutions for the equation x² + 2x + 4 = 0, we can use the quadratic formula, which is:
x = \frac{-b \pm \sqrt{b² – 4ac}}{2a}
In our equation, the coefficients are:
- a = 1 (the coefficient of x²)
- b = 2 (the coefficient of x)
- c = 4 (the constant term)
Now we can substitute these values into the quadratic formula. First, we need to calculate the discriminant:
D = b² – 4ac = 2² – 4(1)(4) = 4 – 16 = -12
Since the discriminant is less than zero (D < 0), this indicates that the equation has two complex solutions. We can now proceed to calculate these solutions using the quadratic formula:
x = \frac{-2 \pm \sqrt{-12}}{2(1)} = \frac{-2 \pm 2i\sqrt{3}}{2} = -1 \pm i\sqrt{3}
Thus, the solutions to the equation x² + 2x + 4 = 0 are:
- x = -1 + i√3
- x = -1 – i√3
These solutions indicate that the parabola represented by the equation does not intersect the x-axis, confirming that the solutions are complex conjugates.