A rectangular patio

A rectangular patio measures 20 ft by 30 ft. By adding x feet to the width and x feet to the length, the area is doubled. Find the new dimensions of the patio.

Correct result:

c =  30 ft
d =  40 ft

Solution:

$$\begin{gathered} a=20\text{ ft } \\ b=30\text{ ft } \\ (a+x)\ast (b+x)=2\ast a\ast b \\ (20+x)\cdot (30+x)=2\cdot 20\cdot 30 \\ {x}^2+50x-600=0 \\ a=1;b=50;c=-600 \\ D=b^2-4ac=50^2-4\cdot 1\cdot (-600)=4900 \\ D>0 \\ {x}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{-50\pm \sqrt{4900}}{2}} \\ {x}_{1,2}={\displaystyle \frac{-50\pm 70}{2}} \\ {x}_{1,2}=-25\pm 35 \\ {x}_1=10 \\ {x}_2=-60 \\ \text{ Factored form of the equation: } \\ (x-10)(x+60)=0 \\ c=a+x_1=20+10=30\text{ ft} \end{gathered}$$

Our quadratic equation calculator calculates it.

$d=b+x_1=30+10=40\text{ ft}$

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