Question

which are the solutions of $x^2 = -13x - 4$?
○ 0, 13
○ 0, -13
○ $\frac{13 - \sqrt{153}}{2}$, $\frac{13 + \sqrt{153}}{2}$
○ $\frac{-13 - \sqrt{153}}{2}$, $\frac{-13 + \sqrt{153}}{2}$

Explanation:

Step1: Rewrite the equation

First, we rewrite the given equation \(x^{2}=-13x - 4\) in standard quadratic form \(ax^{2}+bx + c = 0\). By adding \(13x\) and \(4\) to both sides, we get \(x^{2}+13x + 4=0\). Here, \(a = 1\), \(b = 13\), and \(c = 4\).

Step2: Apply the quadratic formula

The quadratic formula for the solutions of a quadratic equation \(ax^{2}+bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Substituting the values of \(a\), \(b\), and \(c\) into the formula, we have:
First, calculate the discriminant \(D=b^{2}-4ac=(13)^{2}-4\times1\times4 = 169 - 16=153\).
Then, substitute \(a = 1\), \(b = 13\), and \(D = 153\) into the quadratic formula:
\(x=\frac{-13\pm\sqrt{153}}{2\times1}=\frac{-13\pm\sqrt{153}}{2}\)
Which can be written as two solutions: \(\frac{-13-\sqrt{153}}{2}\) and \(\frac{-13+\sqrt{153}}{2}\)

Answer:

\(\frac{-13 - \sqrt{153}}{2},\frac{-13 + \sqrt{153}}{2}\) (the last option among the given choices)