Question 119472
I THINK YOU BEST APPROACH ON WORD PROBLEMS IS TO LAY THEM OUT IN AN ORDERLY  FASHION IDENTIFYING WHAT NEEDS TO BE DETERMINED AS YOU GO.  I HAVE TRIED TO DO THIS BELOW:
Area(A)=Length(l) times Width(w) or A=l*w

Let w=the uniform width of the path. Note that the path is inside the original garden


Note: draw the garden and path and the statement below will be obvious.
Area of garden after path is put in= (20-2w)*(30-2w) =600-100w+4w^2 and this we are told equals 400 ft^2. So:

600-100w+4w^2=400  subtract 400 from both sides

600-400-100w+4w^2=400-400  collect like terms

4w^2-100w+200=0  divide each term by 4

w^2-25w+50=0  quadratic in standard form.  Solve using the quadratic formula:
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 
{{{x = (25 +- sqrt(-25^2-4*1*50 ))/(2*1) }}}
{{{x = (25 +- sqrt( 625-200 ))/(2) }}} 
{{{x = (25 +- sqrt( 425))/(2) }}}  
{{{x = (25 +- 20.6)/(2) }}} 
{{{x = (25 + 20.6)/(2) }}}
{{{x = (45.6)/(2) }}} 
{{{x=22.8}}} ft----------------------too large-- gives a negative value for length and width--- not a correct answer
and
 {{{x = (25 - 20.6)/(2) }}} 
{{{x = (4.4)/(2) }}} 
{{{x=2.2}}} ft----ans
(30-2*2.2)(20-2*2.2)=400
(30-4.4)(20-4.4)=400
25.6*15.6=400

~~~400=400

Hope this helps---ptaylor