Question

find all solutions of the equation $x^2 + 8x + 24 = 0$ by using the quadratic formula.
$x = $

Explanation:

Step1: Identify coefficients

For quadratic equation \(ax^2 + bx + c = 0\), here \(a = 1\), \(b = 8\), \(c = 24\).

Step2: Quadratic formula

Quadratic formula is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\).
Substitute \(a\), \(b\), \(c\): \(x=\frac{-8\pm\sqrt{8^2 - 4\times1\times24}}{2\times1}\).

Step3: Calculate discriminant

Compute \(b^2 - 4ac = 64 - 96=-32\).

Step4: Simplify square root

\(\sqrt{-32}=\sqrt{32}\times\sqrt{-1}=4\sqrt{2}i\) (where \(i\) is imaginary unit).

Step5: Solve for x

\(x=\frac{-8\pm4\sqrt{2}i}{2}=-4\pm2\sqrt{2}i\).

Answer:

\(x = -4 + 2\sqrt{2}i\) or \(x = -4 - 2\sqrt{2}i\)