I have this characteristic equation from a matrix:
$$ 0 = \lambda^3 - \frac{3a^2}{4}\lambda - \frac{1}{4}a^3$$
Where a is some constant.
I have no idea how to go about doing this by hand, I know the answer. I feel like I just missed this in my background of learning mathematics. Any guidance would be appreciated and in general how to solve problems like this.
This can be solved using the cubic formula, $$ax^{3}+bx^{2}+cx+d=0,\ x=\frac{\sqrt[3]{\sqrt{\left(b^{3}-4.5abc+13.5a^{2}d\right)^{2}-\left(b^{2}-3ac\right)^{3}}-\left(b^{3}-4.5abc+13.5a^{2}d\right)}-\sqrt[3]{\sqrt{\left(b^{3}-4.5abc+13.5a^{2}d\right)^{2}-\left(b^{2}-3ac\right)^{3}}+\left(b^{3}-4.5abc+13.5a^{2}d\right)}-b}{3a}$$ which reduces down greatly to give $$λ=a$$