First Principles AI

Solution

\begin{aligned} & \boxed{\text{Let } u = z^2 \text{ to form a quadratic equation:}} \\ \\ & z^4 + 7z^2 - 144 = 0 \implies u^2 + 7u - 144 = 0 \\ \\ & \boxed{\text{Quadratic formula: } u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}} \\ \\ & u = \frac{-7 \pm \sqrt{49 - 4(1)(-144)}}{2(1)} = \frac{-7 \pm \sqrt{49 + 576}}{2} = \frac{-7 \pm \sqrt{625}}{2} = \frac{-7 \pm 25}{2} \\ \\ & u = \frac{-7 + 25}{2} = \frac{18}{2} = 9 \quad \text{or} \quad u = \frac{-7 - 25}{2} = \frac{-32}{2} = -16 \\ \\ & \boxed{\text{Since } u = z^2 \text{, we have } z^2 = 9 \text{ or } z^2 = -16} \\ \\ & z^2 = 9 \implies z = \pm 3 \\ \\ & z^2 = -16 \implies z = \pm 4i \quad (\text{since } \sqrt{-16} = \pm 4i) \\ \\ & \boxed{\text{Therefore, the roots are: } 3, -3, 4i, -4i} \\ \\ & \boxed{\text{Using the factor theorem:}} \\ \\ & z^4 + 7z^2 - 144 = (z - 3)(z + 3)(z - 4i)(z + 4i) \end{aligned}