To remove the $2^{\mathrm{nd}}$ term of the equation ${\mathrm{x}}^4 – 10{\mathrm{x}}^3 + 35{\mathrm{x}}^2 – 50\mathrm{x} + 24 = 0$, diminish the roots by

$\begin{array}{l}f\left(x\right)=x^4-10x^3+35x^2-50x+24=0\\ Letn=4,r=2\\ \therefore f^{(n-r+1)}\left(h\right)=0\\ \Rightarrow f^{(4-2+1)}\left(h\right)=0\Rightarrow f^3\left(h\right)=0\\ f^{\text{'}}\left(x\right)=4x^3-30x^2+70x-50\\ f^{\text{'}\text{'}}\left(x\right)=12x^2-60x+70\\ f^{\text{'}\text{'}\text{'}}\left(x\right)=24x-60\\ \therefore f^{\text{'}\text{'}\text{'}}\left(h\right)=0\\ \Rightarrow 24h-60=0\\ \Rightarrow h=\frac{60}{24}=\frac{5}{2}\end{array}$