To find the solutions to the quadratic equation 2x² + 16x + 50 = 0, we can use the quadratic formula, which is given by:
x = (-b ± √(b² – 4ac)) / 2a
In this equation, a, b, and c are the coefficients from the standard form of a quadratic equation ax² + bx + c = 0.
For our equation:
- a = 2
- b = 16
- c = 50
Now, we can plug these values into the quadratic formula:
1. First, we need to calculate the discriminant (b² – 4ac):
b² = 16² = 256
4ac = 4 * 2 * 50 = 400
Discriminant = 256 – 400 = -144
2. Since the discriminant is negative (-144), it indicates that there are no real solutions; instead, there are two complex solutions.
3. Next, we calculate the solutions using the formula:
x = (-16 ± √(-144)) / (2 * 2)
Simplifying this:
x = (-16 ± 12i) / 4
4. We can now simplify further:
x = -4 ± 3i
Thus, the two solutions to the equation 2x² + 16x + 50 = 0 are:
- x = -4 + 3i
- x = -4 – 3i
In conclusion, the equation has two complex solutions: -4 + 3i and -4 – 3i. These solutions reflect the nature of quadratic equations where the discriminant is negative, indicating the curve does not intersect the x-axis.