Question 1207822
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One formula stating the relationship between the length {{{highlight(cross(l))}}} <U>L</U> and width <U>w</U> of a rectangle 
of “pleasing proportion” is {{{highlight(cross(w))}}} L^2 = w(L + w). How should a 4 {{{highlight(cross(foot))}}} <U>feet</U> by 8 {{{highlight(cross(foot))}}} <U>feet</U> sheet 
of plasterboard be cut so that the result is a rectangle of “pleasing proportion” with a width of 4 feet?
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<pre>
Let's take w= 4 feet and find L from this equation

    L^2 = w*(L + w).


So,  

    L^2 = 4*(L + 4)

    L^2 - 4L - 16 = 0

    {{{L[1,2]}}} = {{{(4 +- sqrt(4^2 - 4*(-16)))/2}}} = {{{(4 +- sqrt(80))/2}}} = {{{2 +- 2*sqrt(5)}}}.


We disregard the negative root and accept the positive one  L = {{{2+2*sqrt(5)}}} = 6.472135955.


So, the cut should be at L = 6.472 feet from the 4 ft edge.    <U>ANSWER</U>
</pre>

Solved.


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By the way, this problem was solved at this forum many years ago (~ 15 years ago) under this link

https://www.algebra.com/algebra/homework/quadratic/Quadratic_Equations.faq.question.669064.html