Answer:
[tex]x =\dfrac 12\left( -5 +\sqrt{13} \right)\\\\ x =\dfrac 12\left( -5 -\sqrt{13} \right)[/tex]
Step-by-step explanation:
[tex]~~~~~x^2 = -5x -3 \\\\\\\implies x^2 +5x =-3\\ \\\implies x^2 + 2 \cdot \dfrac 52 \cdot x + \left( \dfrac 52 \right)^2 = -3 + \left( \dfrac 52 \right)^2\\\\\\\implies \left(x + \dfrac 52 \right)^2 = -3 + \dfrac{25}4\\ \\\\\implies \left(x + \dfrac 52 \right)^2 = \dfrac{13}{4}\\\\\\\implies x+ \dfrac 52 = \pm\sqrt{ \dfrac{13}4}\\\\\\\implies x + \dfrac 52 = \pm\dfrac{\sqrt{13}}{2}\\\\\\\implies x = -\dfrac 52\pm\dfrac{\sqrt{13}}{2}\\\\\\[/tex]
[tex]\implies x =\dfrac 12\left( -5 \pm \sqrt{13} \right)[/tex]
