let

x = the width of the board

y = the length of the board

the perimeter is 28 cm

2(x + y) = 28

x + y = 14 => y = 14 - x

the area is A = x*y

A = x (14 - x)

A = -x^2 + 14x

we need to find the maximum of the parabolic function A(x) = -x^2 + 14x

since the quotient infront of x^2 is negative follows the function has maximum in the vertex equal to

Amax = c - (b^2/4a), where a = -1, b = 14, c=0

Amax = 0 - (14^2/4(-1)) = 49

lets find x such that A(x) = 49

-x^2 + 14x = 49

-x^2 + 14x - 49 = 0

by solving the quadratic equation we find

x = 7 cm

y = 7 cm

the board is a square with side equal to 7 cm.