Answer
20
Work Step by Step
The area of the garden is 16x12.
The area of the entire field is (16+2x)x(12+2x).
Write an equation showing that the area of the garden is 60% of (times) the are of the field.
$16\times12=0.6(16+2x)(12+2x)\longrightarrow$ multiply
$192=0.6(16+2x)(12+2x)\longrightarrow$ divide each side by 0.6
$192\div0.6=0.6(16+2x)(12+2x)\div0.6\longrightarrow$ divide
$320=(16+2x)(12+2x)\longrightarrow$ multiply the binomials
$320=192+56x+4x^2\longrightarrow$ subtract 320 from each side
$320-320=192+56x+4x^2-320\longrightarrow$ subtract
$0=4x^2+56x-128\longrightarrow$ multiply each side by $\frac{1}{4}$
$0\times\frac{1}{4}=\frac{1}{4}(4x^2+56x-128)\longrightarrow$ multiply using the distributive property
$0=x^2+14x-32\longrightarrow$ factor the trinomial; the factors of -32 that sum to 14 are -2 and 16
$0=(x+16)(x-2)\longrightarrow$ one of the 2 factors must be 0
$x+16=0$ or $x-2=0$
$x=-16$ or $x=2$
x cannot be negative, so x=2
The length of the longest side is
$16+2x=16+2(2)=16+4=20.$
