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# complex numbers

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When x = (3 + 5i)/2, find the value of 2x^3 + 2x^2 - 7x + 72.

May 22, 2020

#1
+25648
+2

When

$$x = \dfrac{3 + 5i}{2}$$,

find the value of  $$2x^3 + 2x^2 - 7x + 72$$.

$$\begin{array}{|rcll|} \hline x^2 &=& \left(\dfrac{3 + 5i}{2}\right)^2 \\\\ x^2 &=& \dfrac{(3 + 5i)^2}{4} \\\\ x^2 &=& \dfrac{9+30i-25}{4} \\\\ x^2 &=& \dfrac{-16+30i}{4} \\\\ \mathbf{x^2} &=& \mathbf{\dfrac{-8+15i}{2}} \\ \hline \end{array} \begin{array}{|rcll|} \hline x^3 &=& \dfrac{(-8+15i)}{2}*\dfrac{(3 + 5i)}{2} \\\\ x^3 &=& \dfrac{(-8+15i)(3 + 5i)}{4} \\\\ x^3 &=& \dfrac{-24-40i+45i-75}{4} \\\\ x^3 &=& \dfrac{-99+5i}{4} \\\\ \mathbf{x^3} &=& \mathbf{\dfrac{-99+5i}{4}} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \mathbf{2x^3 + 2x^2 - 7x + 72} &=& 2*\dfrac{(-99+5i)}{4} + 2*\dfrac{(-8+15i)}{2} - 7*\dfrac{(3 + 5i)}{2} + 72 \\\\ &=& \dfrac{-99+5i}{2} + (-8+15i)- \dfrac{7(3 + 5i)}{2} + 72 \\\\ &=& \dfrac{-99+5i}{2} -8+15i- \dfrac{21 + 35i}{2} + 72 \\\\ &=& \dfrac{-99+5i}{2} - \dfrac{21 + 35i}{2} + 64 +15i \\\\ &=& \dfrac{-99+5i-(21 + 35i)}{2} + 64 +15i \\\\ &=& \dfrac{-99+5i-21 - 35i}{2} + 64 +15i \\\\ &=& \dfrac{-120-30i}{2} + 64 +15i \\\\ &=& -60-15i + 64 +15i \\\\ &=& \mathbf{4} \\ \hline \end{array}$$

May 22, 2020