Question 49005
Assuming that the sidewalk will be of uniform width around the lawn, let's call the width of the sidewalk x feet.

With the dimensions given in the problem, the dimensions of the remaining lawn after the sidewalk is constructed will be:
(32-2x) by (24-2x) and its area is given as 425 sq.ft.
So you can write the equation for the area of the new lawn as:
{{{(32-2x)(24-2x) = 425}}} Simplify this and solve it for x, the width of the sidewalk.
{{{768 - 112x - 4x^2 = 425}}} Subtract 425 from both sides of the equation.
{{{4x^2 - 112x + 343 = 0}}} Solve this quadratic for x by factoring.
{{{(2x-49)(2x-7) = 0}}} Apply the zero product principle.
{{{2x-49 = 0}}} and/or {{{2x-7 = 0}}}
If {{{2x-49 = 0}}} then {{{2x = 49}}} and {{{x = 49/2}}}
If {{{2x-7 = 0}}} then {{{2x = 7}}} and {{{x = 7/2}}}

As you would expect, we have two solutions to this quadratic equation...but only one of them is meaningful in terms of the width of the sidewalk. Let's look at them one-at-a-time.

{{{x = 49/2}}} = 24.5 feet This is not meaningful because the original lawn is only 24 feet wide and if you built a sidewalk this wide, you would have no lawn left. So, discard this solution.

{{{x = 7/2}}} = 3.5 feet. This solution works.  Let's check!

{{{(32-2x)(24-2x) = 425}}} Substituting x = 3.5
{{{(32-7)(24-7) = 425}}} Simplify.
{{{(25)(17) = 425}}}
{{{425 = 425}}} It checks!