Question

final review
score: 50/100 answered: 10/17
question 11
find all solutions of the equation $x^2 + 6x + 59 = 0$ by using the quadratic formula.
$x = $
hint: reduce radicals. express answers as two numbers separated by a comma. no decimal approximations.

Explanation:

Step1: Recall Quadratic Formula

For a quadratic equation \(ax^2 + bx + c = 0\), the solutions are given by \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 1\), \(b = 6\), \(c = 59\).

Step2: Calculate Discriminant

Compute \(b^2-4ac=(6)^2 - 4(1)(59)=36 - 236=-200\).

Step3: Substitute into Formula

Substitute \(a\), \(b\), and the discriminant into the formula: \(x=\frac{-6\pm\sqrt{-200}}{2(1)}\). Simplify \(\sqrt{-200}=\sqrt{100\times(- 2)} = 10i\sqrt{2}\)? Wait, no: \(\sqrt{-200}=\sqrt{200}\times\sqrt{-1}=10\sqrt{2}i\). So \(x=\frac{-6\pm10\sqrt{2}i}{2}=-3\pm5\sqrt{2}i\).

Answer:

\(-3 + 5\sqrt{2}i, -3 - 5\sqrt{2}i\)