#"let the length "=l#
#"then width "=l-1/2larrcolor(blue)" half metre shorter"#
#"area "="length"xx"width"=l(l-1/2)=l^2-1/2l#
#"we are given the area "=33#
#"equating the 2 gives"#
#l^2-1/2l=33larrcolor(blue)"multiply through by 2"#
#rArr2l^2-l=66larrcolor(blue)"express in standard form"#
#rArr2l^2-l-66=0larrcolor(blue)"in standard form"#
#"using the a-c method to factor the quadratic"#
#"the factors of the product "2xx-66=-132#
#"which sum to - 1 are - 12 and + 11"#
#"split the middle term using these factors"#
#2l^2-12l+11l-66=0larrcolor(blue)"factor by grouping"#
#color(red)(2l)(l-6)color(red)(+11)(l-6)#
#"take out the "color(blue)"common factor "(l-6)#
#(l-6)(color(red)(2l+11))#
#rArr(l-6)(2l+11)=0#
#"equate each factor to zero and solve for "l#
#l-6=0rArrl=6#
#2l+11=0rArrl=-11/2#
#"but "l>0rArrl=6#
#"and width "=l-1/2=6-1/2=5 1/2#
#color(blue)"As a check"#
#"area "=6xx5 1/2=33color(white)(x)✔︎#
