Question 1194635
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Finding the width of the walkway (which SEEMS to be the objective) is much easier using common sense than with formal algebra.<br>
The sections of the walkway are two sections 500 by x meters, two sections 250 by x meters, and four section x by x meters.  With a total walkway area of 1504 meters, that gives us<br>
{{{2(500x)+2(250x)+4x^2=1504}}}
{{{1500x+4x^2=1504}}}<br>
Simple observation shows that x=1.<br>
But the problem doesn't ask us to find the width of the walkway; it asks us what the roots are of the "working equation".<br>
If we use the preceding discussion to form the "working equation", we have<br>
{{{4x^2+1500x-1504=0}}}
{{{x^2+375x-376=0}}}
{{{(x+376)(x-1)=0}}}<br>
ANSWER: The roots are x=-376 and x=1<br>
A more standard way of setting up the problem leads to the same "working equation" by a different path.<br>
The area of the walkway is the area of (pool plus walkway), minus the area of the pool:<br>
{{{(500+2x)(250+2x)-(500)(250)=1504}}}
{{{(500)(250)+1500x+4x^2-(500)(250)=1504}}}
{{{4x^2+1500x-1504=0}}}<br>