Question

solve by using the quadratic form
$x^2 = -8x - 19$
separate your answers with co
express the solution set in exa

Explanation:

Step1: Rewrite in standard form

First, we rewrite the equation \(x^{2} = - 8x - 19\) in the standard quadratic form \(ax^{2}+bx + c=0\). By adding \(8x\) and \(19\) to both sides, we get \(x^{2}+8x + 19=0\). Here, \(a = 1\), \(b = 8\), and \(c=19\).

Step2: Apply quadratic formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Substitute \(a = 1\), \(b = 8\), and \(c = 19\) into the formula. First, calculate the discriminant \(\Delta=b^{2}-4ac=(8)^{2}-4\times1\times19=64 - 76=- 12\). Then, \(x=\frac{-8\pm\sqrt{-12}}{2\times1}\). Since \(\sqrt{-12}=\sqrt{12}\times\sqrt{-1} = 2\sqrt{3}i\) (where \(i\) is the imaginary unit), we have \(x=\frac{-8\pm2\sqrt{3}i}{2}\).

Step3: Simplify the expression

Simplify \(\frac{-8\pm2\sqrt{3}i}{2}\) by dividing each term in the numerator by \(2\). We get \(x=-4\pm\sqrt{3}i\).

Answer:

\(x=-4 + \sqrt{3}i,-4-\sqrt{3}i\)