Question

solve the equation.\\( x^2 + 3x + 6 = 0 \\)\\( x = -\frac{?}{} pm \frac{sqrt{}}{i} \\)give your answer as a complex number.

Explanation:

Step1: Recall Quadratic Formula

For a quadratic equation \(ax^2 + bx + c = 0\), the solutions are given by \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). For complex solutions, when \(b^2-4ac<0\), we can write \(\sqrt{b^2 - 4ac}=\sqrt{-(4ac - b^2)}=i\sqrt{4ac - b^2}\).

Given the equation \(x^2+3x + 6 = 0\), we have \(a = 1\), \(b = 3\), and \(c = 6\).

Step2: Calculate the discriminant part

First, calculate \(b^2-4ac\):
\(b^2-4ac=(3)^2-4\times1\times6=9 - 24=- 15\)

Since the discriminant is negative, we rewrite the square - root part as \(i\sqrt{15}\) (because \(\sqrt{-15}=i\sqrt{15}\)).

Now, apply the quadratic formula:
\(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}=\frac{-3\pm\sqrt{-15}}{2}=\frac{-3\pm i\sqrt{15}}{2}=-\frac{3}{2}\pm\frac{\sqrt{15}}{2}i\)

Answer:

For the first (green) box, the numerator is \(3\), the denominator of the first fraction is \(2\), the number under the square - root in the second fraction is \(15\), and the denominator of the second fraction is \(2\). So the filled - in form is \(x =-\frac{3}{2}\pm\frac{\sqrt{15}}{2}i\)