Golden ratio

Divide a line segment of length 871 cm into two parts so that the ratio of the shorter part to the longer part equals the ratio of the longer part to the whole length.

Final Answer:

x₁ =  332.69 cm
x₂ =  538.31 cm

Step-by-step explanation:

$$\begin{gathered} \frac{x_1}{x_2}=\frac{x_2}{871} \\ {x}_1+x_2=871 \\ {x}_1\cdot 871=x_2^2 \\ {x}_1\cdot 871=(871-x_1{)}^2 \\ a\cdot 871=(871-a{)}^2 \\ a\cdot 871=(871-a{)}^2 \\ -a^2+2613a-758641=0 \\ {a}^2-2613a+758641=0 \\ p=1;q=-2613;r=758641 \\ D=q^2-4pr=2613^2-4\cdot 1\cdot 758641=3793205 \\ D>0 \\ {a}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{2613\pm \sqrt{3793205}}{2}=\frac{2613\pm 871\sqrt{5}}{2} \\ {a}_{1,2}=1306.5\pm 973.807604 \\ {a}_1=2280.307604201 \\ {a}_2=332.692395799 \\ {x}_1=a_2=332.6924=332.69\text{cm} \end{gathered}$$

Our quadratic equation calculator calculates it.

$$\begin{gathered} x_2=871-x_1=871-332.6924\doteq 538.3076 \\ \text{ Verifying Solution: } \\ k=(1+\sqrt{5})/2\doteq 1.618 \\ {X}_2=x_1\cdot k=332.6924\cdot 1.618\doteq 538.3076\text{cm} \\ {x}_2=X_2=538.31\text{cm} \end{gathered}$$

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