Question

1. what are the solutions of $x^2 + 2x + 6 = 0$?

Explanation:

Step1: Identify the quadratic formula

For a quadratic equation \(ax^{2}+bx + c = 0\), the solutions are given by \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
Here, \(a = 1\), \(b = 2\), and \(c = 6\).

Step2: Calculate the discriminant

The discriminant is \(D=b^{2}-4ac\).
Substitute \(a = 1\), \(b = 2\), \(c = 6\) into the discriminant formula:
\(D=(2)^{2}-4\times1\times6\)
\(D = 4 - 24\)
\(D=-20\)

Step3: Substitute into the quadratic formula

Since the discriminant is negative, we will have complex solutions. Recall that \(\sqrt{-1}=i\), so \(\sqrt{-20}=\sqrt{20}\times\sqrt{-1}=2\sqrt{5}i\).
Now, substitute \(a = 1\), \(b = 2\), and \(D=-20\) into the quadratic formula:
\(x=\frac{-2\pm\sqrt{-20}}{2\times1}\)
\(x=\frac{-2\pm2\sqrt{5}i}{2}\)

Step4: Simplify the expression

Divide each term in the numerator by 2:
\(x=-1\pm\sqrt{5}i\)

Answer:

The solutions are \(x=-1+\sqrt{5}i\) and \(x=-1-\sqrt{5}i\)