Question 320785
Let the width of the path be x feet.
The plan is to express the area of the garden with the border in terms of x, then subtract the area of just the garden.  This will result in a quadratic equation in x which you solve by any of the appropriate methods available to you.
The area of the garden with border {{{A[1]}}} can be expressed as:
{{{A[1] = (30+2x)(20+2x)}}} Simplify this to:
{{{A[1] = 4x^2+100x+600}}}
The area of just the garden {{{A[2]}}} is:
{{{A[2] = (20)*(30)}}}
{{{A[2] = 600}}}sq.ft.
Now subtract {{{A[2]}}} from {{{A[1]}}}.
{{{A[1] - A[2] = 4x^2+100x}}} This is the area of the border of width x ft. which is given as 336 sq.ft., so...
{{{4x^2+100x = 336}}} Subtract 336 from both sides.
{{{4x^2+100x-336 = 0}}} Simplify by factoring out a 4.
{{{4(x^2+25x-84) = 0}}} ...and from the zero product rule, you get that:
{{{x^2+25x-84 = 0}}} You can solve this by factoring and you should get:
{{{x = 3}}} or {{{x = -28}}} 
The width cannot be a negative quantity, so the width must be 3 feet.