Question

Transform the equation $x^3-\frac{x}{4}-\frac{3}{4}=0$ into one whose roots exceed by $\frac{3}{2}$ the corresponding roots of the given equation.

Answer 1

$f\left(x\right)=x^3-\frac{x}{4}-\frac{3}{4}=0⟹f\left(x\right)=4x^3-x-3=0$

$y=x+\frac{3}{2}⟹x=y-\frac{3}{2}$

$⟹f\left(y\right)=4{(y-\frac{3}{2})}^3-(x-\frac{3}{2})-3=0$

$⟹f\left(y\right)=4y^3-18y^2+27y-\frac{27}{2}–y+\frac{3}{2}-3=0$

$⟹f\left(y\right)=4y^3-18y^2+26y-15=0$