+2864 \(\frac{l+w}{l}=\frac{l}{w}\)
Find \(\frac{l}{w}\), your answer should be the golden ratio that is a fraction with radicals in simplest form.
(BTW, the answer is definitely NOT \(\frac{l+w}{l}\))
This is a fun math problem I solved yesterday night.
+2864 You incorrectly multiplied by lw,
it should be lw+w^2=l^2
You also CAN'T subtract l^2
+26393 Easy Golden Ratio Quadratic
\(\dfrac{l+w}{l}=\dfrac{l}{w}\)
I assume:
\(\text{Let }\textbf{Golden Ratio} =\varphi\)
\(\begin{array}{|rcll|} \hline \dfrac{l+w}{l} &=& \dfrac{l}{w} \\\\ \dfrac{l}{l}+ \dfrac{w}{l} &=& \dfrac{l}{w} \\\\ 1 + \dfrac{w}{l} &=& \dfrac{l}{w} \quad | \quad \dfrac{l}{w} = \varphi,\ \dfrac{w}{l} = \dfrac{1}{\varphi} \\\\ 1 + \dfrac{1}{\varphi} &=& \varphi \quad |\quad \cdot \varphi \\\\ \varphi + 1 &=& \varphi^2 \\ \varphi^2 -\varphi - 1 &=& 0 \\\\ \varphi &=& \dfrac{1\pm \sqrt{1-4\cdot (-1) }}{2} \\ \varphi &=& \dfrac{1\pm \sqrt{5}}{2} \\\\ \mathbf{\varphi} &\mathbf{=}& \mathbf{\dfrac{1+ \sqrt{5}}{2}} \\ \hline \end{array}\)
