Solving the Equation x² + 2x + 6 = 0
To solve the quadratic equation x² + 2x + 6 = 0, we can use several methods. However, in this case, the quadratic formula is likely the most effective method because it provides a solution for any quadratic equation.
Step 1: Identifying Coefficients
The general form of a quadratic equation is: ax² + bx + c = 0, where:
- a is the coefficient of x²,
- b is the coefficient of x,
- c is the constant term.
For our equation, we have:
- a = 1,
- b = 2,
- c = 6.
Step 2: Applying the Quadratic Formula
The quadratic formula is given by:
x = (-b ± √(b² – 4ac)) / (2a).
Now, we will substitute the coefficients into the formula:
- Calculate b² – 4ac:
- 2² – 4 * 1 * 6 = 4 – 24 = -20.
Since the discriminant (b² – 4ac) is negative (-20), we will have complex (imaginary) solutions.
Step 3: Finding the Roots
Next, we plug in the values into the quadratic formula:
- Calculate x = (-2 ± √(-20)) / (2 * 1).
We can simplify this further:
- x = (-2 ± √(20)i) / 2 (since √(-1) = i, the imaginary unit)
- √20 can be simplified to 2√5, so we have:
- x = (-2 ± 2√5i) / 2 => x = -1 ± √5i.
Final Answer
Thus, the solutions to the equation x² + 2x + 6 = 0 are:
- x = -1 + √5i
- x = -1 – √5i
These solutions indicate that the graph of the equation does not intersect the x-axis, confirming that the roots are complex numbers.