Golden ratio

Divide the line of length 871 cm into two sections so that the ratio of shorter to greater is the same as the ratio of the greater section to the whole length of the line.

Correct 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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