Question 103664
let
{{{perimeter = P}}}

{{{area = A}}}

{{{sides}}}: {{{a, and b}}}

{{{P = 2(a+b)}}}
since {{{P = 22cm}}}

{{{22 cm = 2(a + b)}}}   divide both sides by {{{2}}}
{{{11cm = a + b}}}   = > {{{a = 11 - b}}}

now
{{{A = a*b}}}

since  {{{A = 24 cm^2}}} and {{{a = 11 - b}}}
we have:

{{{24 cm^2 = (11 - b)*b }}} 

{{{24 cm^2 = 11b - b^2}}} 

{{{b^2 - 11b + 24 cm^2}}} 
{{{b(1,2)=(11 +- sqrt ((-11)^2 -4*1*24)) / (2*1)}}}

{{{b(1,2)=(11 +- sqrt (121 -96)) / (2)}}}

{{{b(1,2)=(11 +- sqrt (25)) / 2}}}

{{{b(1,2)=(11 +- 5) / 2}}}

{{{b(1)=(11 + 5 )/ 2}}} = > {{{b1 = 8}}}


{{{b(2)=(11 -5) / 2}}}  = > {{{b2 = 3 }}}


since {{{a = 11 - b}}}

we have:

{{{a = 11 - b1}}} => {{{a1 = 11 - 8}}}  or {{{a1 = 3}}}
{{{a = 11 - b2}}} => {{{a1 = 11 - 3}}}  or {{{a1 = 8}}}

=>
{{{a = 8}}}{{{cm}}}
{{{b= 3 }}}{{{cm}}}         or vice versa

check:
{{{P= 2(a+b)}}}
{{{P = 2(8+3)}}}
{{{P = 2*11}}}
{{{P = 22cm}}}

{{{A= 8*3cm^2}}}
{{{A = 24cm^2}}}