Question

solve the equation in the complex number system.
$x^2 + x + 6 = 0$
the solution set is \left\\{ -\frac{1}{2} + \frac{i\sqrt{23}}{2}, -\frac{1}{2} - \frac{i\sqrt{23}}{2} \
ight\\}
(simplify your answer. type an exact answer, using radicals and $i$ as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)

Explanation:

Step1: Identify the formula

To solve the quadratic equation \(ax^2 + bx + c = 0\) in the complex number system, we use the quadratic formula \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). For the equation \(x^2+x + 6=0\), we have \(a = 1\), \(b = 1\), and \(c = 6\).

Step2: Calculate the discriminant

First, calculate the discriminant \(D=b^2-4ac\). Substituting the values, we get \(D=(1)^2-4\times1\times6=1 - 24=- 23\).

Step3: Apply the quadratic formula

Now, substitute \(a\), \(b\), and \(D\) into the quadratic formula. Since \(\sqrt{-23}=\sqrt{23}\times\sqrt{-1}=i\sqrt{23}\) (where \(i\) is the imaginary unit), we have:

$$
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Answer:

\(-\frac{1}{2}+\frac{i\sqrt{23}}{2},-\frac{1}{2}-\frac{i\sqrt{23}}{2}\)