Question

if X equal to 4 by 3 minus root 5 find the value of x + 4 by x

Answer 1

Dear Student,

$$\begin{gathered} Accordingtoyourprovideddetails \\ x=\frac{4}{3-\sqrt{5}} \\ so, \\ =x+\frac{4}{x} \\ =\frac{4}{3-\sqrt{5}}+\frac{4}{\frac{4}{3-\sqrt{5}}} \\ =\frac{4}{3-\sqrt{5}}+3-\sqrt{5} \\ =\frac{4-{\left(3-\sqrt{5}\right)}^2}{3-\sqrt{5}} \\ =\frac{4-\left(9+5-6\sqrt{5}\right)}{3-\sqrt{5}}\left[{\left(a-b\right)}^2=a^2+b^2-2ab\right] \\ =\frac{4-\left(14-6\sqrt{5}\right)}{3-\sqrt{5}} \\ =\frac{-10+6\sqrt{5}}{3-\sqrt{5}}\times \frac{3+\sqrt{5}}{3+\sqrt{5}} \\ =\frac{\left(-10+6\sqrt{5}\right)\left(3+\sqrt{5}\right)}{3^2-{\sqrt{5}}^2}\left[\left(a+b\right)\left(a-b\right)=a^2-b^2\right] \\ =\frac{-30+18\sqrt{5}-10\sqrt{5}+30}{9-5} \\ =\frac{18\sqrt{5}-10\sqrt{5}}{4} \\ =\frac{\sqrt{5}\left(18-10\right)}{4} \\ =\frac{8\sqrt{5}}{4} \\ =\mathbf{2}\sqrt{\mathbf{5}} \end{gathered}$$
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