Question 1179483
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Note in the solution from the other tutor how the standard algebraic solution involves factoring the quadratic equation<br>
{{{x^2+2x-728=0}}}<br>
Few people are going to look at that and see that the factorization is<br>
{{{(x+28)(x-26)=0}}}<br>
In their solution, they solved the equation using the quadratic formula.  That always works; but in this case it still involved working with large numbers.<br>
This is an example of a problem where a bit of insight (or perhaps a lot of experience) can make solving the problem much easier.<br>
Here is what I saw almost immediately on reading the problem: we have two numbers that differ by 2 whose product is 728.  That is two numbers that are very close together, so the 728 is very close to a perfect square.  So instead of calling the numbers x and x+2....<br>
let x-1 be the smaller number (number of trees per row)
let x+1 be the larger number (number of rows of trees)<br>
Then the given information yields an equation that is solved relatively easily:<br>
{{{(x-1)(x+1)=728}}}
{{{x^2-1=728}}}
{{{x^2=729}}}
{{{x=27}}}<br>
ANSWER: The number of trees in each row is x-1=26<br>