Question 125683
the width {{{W}}} of a piece is {{{4 m}}} less than is length {{{L}}}:

{{{W = L - 4m}}}

 the aria {{{A}}} in the room is {{{165 m^2}}}

 find the length and breadth of this piece

given:

{{{W = L - 4m}}}

{{{A = 165 m^2}}} 

since {{{A = L*W}}}, we will have:

{{{165 m^2= L(L - 4m)}}} 

{{{165 m^2= L^2 - 4m*L}}} 

{{{0= L^2 - 4m*L - 165 m^2}}} 

or

{{{ L^2 - 4m*L - 165 m^2 = 0}}} .....use quadratic formula to solve for {{{L}}}



{{{x[1,2] = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 


{{{x[1,2] = (-(-4m) +- sqrt( (-4m)^2-4*1*(- 165 m^2) ))/(2*1) }}} 

{{{x[1,2] = (4m +- sqrt( 16m^2 + 660m^2 ))/2}}} 

{{{x[1,2] = (4m +- sqrt( 676m^2 ))/2}}} 

{{{x[1,2] = (4m +- 26m)/2}}} 


{{{x[1] = (4m + 26m)/2}}} 

{{{x[1] =  30m/2}}} 

{{{x[1] =  15m}}} 


{{{x[2] = (4m - 26m)/2}}} 

{{{x[2] =  -22m/2}}} 

{{{x[2] =  -11m}}} ....since this solution is a negative number, we will exclude this solution because the lenght coud be only positive number

so, the lenght is {{{L = 15m}}}

then the width is 

{{{W = L - 4m}}}....plug in {{{L}}}

{{{W = 15m - 4m}}}

{{{W = 11m }}}


check the area:

{{{A = L*W}}}
{{{A = 15m*11m}}}
{{{A = 165m^2}}}