Question 120184
First, the area of the garden {{{A[g]}}} is:
{{{A[g] = 6*4}}}={{{24}}}sq.m
The total area {{{A[t]}}} of the garden plus the surrounding border is:
{{{A[t] = (6+2x)(4+2x)}}} where x is the width of the uniform border.
Since the area of the border is to equal the area of the garden, we need to find the area of the border alone, which is:
{{{A[t] - A[g]}}} and this is to equal the area of the garden {{{A[g]}}}, so...
{{{A[t]-A[g] = A[g]}}} or...
{{{A[t]-2A[g] = 0}}} Let's find {{{A[t]}}}
{{{A[t] = (6+2x)(4+2x)}}}
{{{A[t] = 24+20x+4x^2}}} and the area of the garden, {{{A[g] = 6*4}}} or:
{{{A[g] = 24}}} Now we'll subtract:
{{{A[t]-2A[g] = 0}}}
{{{4x^2+20x+24-2(24) = 0}}}
{{{4x^2+20x-24 = 0}}} Factor out a 4 to simplify calculations.
{{{4(x^2+5x-6) = 0}}} so that...
{{{x^2+5x-6 = 0}}} Solve this quadratic equation by factoring.
{{{(x-1)(x+6) = 0}}} Apply the zero product principle:
{{{x-1 = 0}}} or {{{x+6 = 0}}}
If {{{x-1 = 0}}} then {{{x = 1}}} or 
If {{{x+6 = 0}}} then {{{x = -6}}}
So the two solutions to the quadratic are:
{{{x = 1}}} This is a valid solution because the width, x, has to be a positive value.
{{{x = -6}}} This solution is not valid as the width, x, cannot be a negative value.
The border is 1 meter wide.