In which of the two equations is the sum of the squares of the roots greater

a) $4 x^{3}-18 x^{2}+24 x=8, \quad 4 x^{3}-18 x^{2}+24 x=9$

b) $4 x^{3}-18 x^{2}+24 x=11, \quad 4 x^{3}-18 x^{2}+24 x=12 ?$
According to Vieta's formulas, the sum of the squares of the roots of each of the four equations, including complex roots, is $\left({ }^{18} /\right)^{2}-2 \cdot 6=8,25$. Let's find the number of real roots of each equation.

The derivative of the function $f(x)=4 x^{3}-18 x^{2}+24 x$ is $12\left(x^{2}-3 x+2\right)$ and equals zero at points $x_{1}=1$ and $x_{2}=$ 2. At these points, $f(1)=10$ and $f(2)=8$.

a) Both equations have three roots (the second equation has a multiple root $x_{2}=2$). Therefore, the sums of their squares are the same.
b) Both equations have one root, and due to the increasing nature of the function $f$ on the interval $(2,+\infty)$, the root of the second equation is greater.

## Answer

a) The same; b) the root of the second equation is greater.