Question

solve $x^2 + 6x + 15 = 0$ using the quadratic formula.
$x = \square$ and $x = \square$

Explanation:

Step1: Recall Quadratic Formula

For a quadratic equation \(ax^2 + bx + c = 0\), the quadratic formula is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\).

Step2: Identify \(a\), \(b\), \(c\)

In the equation \(x^2 + 6x + 15 = 0\), we have \(a = 1\), \(b = 6\), \(c = 15\).

Step3: Calculate the discriminant (\(b^2 - 4ac\))

Substitute the values of \(a\), \(b\), \(c\) into the discriminant formula:
\(b^2 - 4ac=(6)^2 - 4(1)(15)=36 - 60=-24\)

Step4: Substitute into Quadratic Formula

Now substitute \(a = 1\), \(b = 6\), and \(b^2 - 4ac=-24\) into the quadratic formula:
\(x=\frac{-6\pm\sqrt{-24}}{2(1)}\)
Simplify \(\sqrt{-24}\) as \(\sqrt{24}\times\sqrt{-1}=2\sqrt{6}i\) (where \(i\) is the imaginary unit, \(i^2=-1\)).
So, \(x=\frac{-6\pm2\sqrt{6}i}{2}\)

Step5: Simplify the expression

Divide each term in the numerator by 2:
\(x=-3\pm\sqrt{6}i\)

Answer:

\(x = -3 + \sqrt{6}i\) and \(x = -3 - \sqrt{6}i\)