((2+5\sqrt(5)))/(4)=(1)/(2)*(3+2\sqrt(5))*x*((1)/(2))
You asked:
Solve the equation \(\frac{2 + 5 \cdot \sqrt{5}}{4} = \frac{1}{2} \cdot \left(3 + 2 \cdot \sqrt{5}\right) \cdot x \cdot \frac{1}{2}\) for the variable \(x\).
MathBot Answer:
The solution is: \[x = \frac{2 + 5 \sqrt{5}}{3 + 2 \sqrt{5}} \approx 1.763932\]
\[\begin{aligned}\frac{1}{2} + \frac{5 \sqrt{5}}{4} &= x \left(\frac{3}{4} + \frac{\sqrt{5}}{2}\right)\\x \left(- \frac{\sqrt{5}}{2} - \frac{3}{4}\right) &= - \frac{5 \sqrt{5}}{4} - \frac{1}{2}\\x &= \frac{- \frac{5 \sqrt{5}}{4} - \frac{1}{2}}{- \frac{\sqrt{5}}{2} - \frac{3}{4}}\end{aligned}\]