Question 85444
1.  x = first integer
    x + 1 = next consecutive integer

{{{x^2}}} + {{{(x + 1)^2}}} = 85
{{{x^2}}} + {{{x^2}}} + 2x + 1 = 85
{{{2x^2}}} + 2x + 1 = 85
{{{2x^2}}} + 2x - 84 = 0
{{{x^2}}} + x - 42 = 0
(x + 7)(x - 6) = 0
x + 7 = 0 ; x - 6 = 0
x = -7 ; x = 6
x = -7 , x + 1 = -6 is one pair of solutions
x = 6 , x + 1 = 7 is another pair of solutions

2. Let x = width of the path
Garden area = 20(30) = 600
Remaining Garden area = (30 - 2x)(20 - 2x) = 400
600 - 100x + {{{4x^2}}} = 400
{{{4x^2}}} - 100x + 600 = 400
{{{4x^2}}} - 100x + 200 = 0
{{{x^2}}} - 25x + 50 = 0
{{{x = (25 +- sqrt( (-25)^2-4*1*50 ))/(2*1) }}} =
{{{(25 +- sqrt( 625-200 ))/2}}} =
{{{(25 +- sqrt(425))/2}}} =
{{{(25 +- 21.62)/2}}} =
{{{46.62/2}}} , {{{3.38/2}}} =
23.31 , 1.69
23.31 is not feasible since the whole width is only 20 ft, so the answer is 1.69 ft.