Answer :
From the given informatio, we can draw the following picture:
where x denote the shorter side of the rectangle and x+6 the longest one. Since the area of the rectangle is equal to the base times the height, we have
[tex]A=(x+6)x[/tex]Since the area is 520 square centimeters, we have
[tex]520=(x+6)x[/tex]Then, by distributing the variable x into the parentheses, we have the following equation:
[tex]x^2+6x=520[/tex]or equivalently,
[tex]x^2+6x-520=0[/tex]So, we can to apply the quadratic formula:
[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]coming from the quadratic polynomial:
[tex]ax^2+bx+c=0[/tex]By comparing this polynomial with ours, we can note that
[tex]\begin{gathered} a=1 \\ b=6 \\ c=-520 \end{gathered}[/tex]so, we get
[tex]x=\frac{-6\pm\sqrt{6^2-4(1)(-520)}}{2}[/tex]which gives
[tex]\begin{gathered} x=\frac{-6\pm\sqrt{2116}}{2} \\ x=\frac{-6\pm46}{2} \end{gathered}[/tex]Then, we have 2 possible solutions:
[tex]\begin{gathered} x=\frac{-6+46}{2}=\frac{40}{2}=20 \\ and \\ x=\frac{-6-46}{2}=\frac{-52}{2}=-26 \end{gathered}[/tex]However, since the dimension are always positive numbers, the searched side measure 20 cm. In other words, the shorter side measures 20cm and the longest one 20+6=26 cm. Therefore, the answer is: 26 centimeters.
