Rectangle - sides

A rectangle has an area 340 cm². The length of the shorter side is 3 cm fewer than the length of the longer side. What is the perimeter of a rectangle?

Final Answer:

o =  74 cm

Step-by-step explanation:

$$\begin{gathered} S=340{\text{cm}}^2 \\ ab=S \\ b=a+3 \\ a(a+3)=S \\ {a}^2+3\cdot a-S=0 \\ {a}^2+3\cdot a-340=0 \\ {a}^2+3a-340=0 \\ p=1;q=3;r=-340 \\ D=q^2-4pr=3^2-4\cdot 1\cdot (-340)=1369 \\ D>0 \\ {a}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{-3\pm \sqrt{1369}}{2} \\ {a}_{1,2}=\frac{-3\pm 37}{2} \\ {a}_{1,2}=-1.5\pm 18.5 \\ {a}_1=17 \\ {a}_2=-20 \\ a=a_1=17\text{cm} \\ b=a+3=17+3=20\text{cm} \\ o=2\cdot (a+b)=2\cdot (17+20)=74=74\text{cm} \\ \text{ Verifying Solution: } \\ {S}_2=a\cdot b=17\cdot 20=340{\text{cm}}^2 \end{gathered}$$

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