Two gardens

The flower garden has a square shape. The new garden has a rectangular shape; one dimension is 8 m smaller, and the other is twice as large as in a square garden. What were the original and new garden dimensions if both gardens' areas were the same?

Correct answer:

a =  16 m
x =  8 m
y =  32 m

Step-by-step explanation:

$$\begin{gathered} S_1=S_2 \\ {S}_1=a^2 \\ {S}_2=x\cdot y \\ x=a-8 \\ y=2a \\ {a}^2=x\cdot y \\ {a}^2=(a-8)\cdot 2a \\ {a}^2=(a-8)\cdot 2a \\ -a^2+16a=0 \\ {a}^2-16a=0 \\ p=1;q=-16;r=0 \\ D=q^2-4pr=16^2-4\cdot 1\cdot 0=256 \\ D>0 \\ {a}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{16\pm \sqrt{256}}{2} \\ {a}_{1,2}=\frac{16\pm 16}{2} \\ {a}_{1,2}=8\pm 8 \\ {a}_1=16 \\ {a}_2=0 \\ a=a_1=16=16\text{m} \end{gathered}$$

Our quadratic equation calculator calculates it.

$x=a-8=16-8=8\text{m}$

$$\begin{gathered} y=2\cdot a=2\cdot 16=32=32\text{m} \\ \text{ Verifying Solution: } \\ {S}_1=a^2=16^2=256{\text{m}}^2 \\ {S}_2=x\cdot y=8\cdot 32=256{\text{m}}^2 \end{gathered}$$

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