Solving the Quadratic Equation x² + 6x + 18
To solve the quadratic equation x² + 6x + 18 = 0, we can employ the quadratic formula, which is given by:
x = (-b ± √(b² – 4ac)) / 2a
In this formula, a, b, and c are coefficients from the equation in the standard form ax² + bx + c = 0. For this particular equation:
- a = 1
- b = 6
- c = 18
Step 1: Calculate the Discriminant
The first step is to calculate the discriminant, D = b² – 4ac. This value will help us determine the nature of the roots:
D = 6² – 4(1)(18)
D = 36 – 72
D = -36
Since the discriminant is negative (D < 0), this indicates that there are no real solutions; instead, there will be two complex solutions.
Step 2: Apply the Quadratic Formula
Next, we will substitute a, b, and c into the quadratic formula:
x = (-6 ± √(-36)) / (2 * 1)
Since the discriminant is negative, we can express it as follows:
x = (-6 ± 6i) / 2
Where i is the imaginary unit.
Step 3: Simplify the Expression
Now, simplifying this gives us:
x = -3 ± 3i
Thus, the two complex solutions are:
- x = -3 + 3i
- x = -3 – 3i
Conclusion
The quadratic equation x² + 6x + 18 = 0 has two complex solutions: x = -3 + 3i and x = -3 – 3i. Understanding how to work with complex numbers is essential when encountering negative discriminants in quadratic equations. If you have any further questions on this topic, feel free to ask!