Flos/Problems/Problem 2

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Problem from Flos by Leonardo Fibonacci

Solve the cubic:
$x^3 + 2 x^2 + 10 x = 20$


Solution

$1 \cdotp 3688081075 \dots$


Proof

Express the equation as:

$x^3 + 2 x^2 + 10 x - 20 = 0$

We apply Cardano's Formula, with $a = 1$, $b = 2$, $c = 10$ and $d = -20$.

Thus:

\(\ds Q\) \(=\) \(\ds \dfrac {3 a c - b^2} {9 a^2}\)
\(\ds \) \(=\) \(\ds \dfrac {3 \times 1 \times 10 - 2^2} {9 \times 1^2}\)
\(\ds \) \(=\) \(\ds \dfrac {26} 9\)
\(\ds R\) \(=\) \(\ds \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}\)
\(\ds \) \(=\) \(\ds \dfrac {9 \times 1 \times 2 \times 10 - 27 \times 1^2 \times \paren {-20} - 2 \times 2^3} {54 \times 1^3}\)
\(\ds \) \(=\) \(\ds \dfrac {180 + 540 - 16} {54}\)
\(\ds \) \(=\) \(\ds \dfrac {352} {27}\)


Then:

\(\ds S\) \(=\) \(\ds \sqrt [3] {R + \sqrt {Q^3 + R^2} }\)
\(\ds \) \(=\) \(\ds \sqrt [3] {\frac {352} {27} + \sqrt {\paren {\frac {26} 9}^3 + \paren {\frac {352} {27} }^2} }\)
\(\ds \) \(=\) \(\ds \sqrt [3] {\frac {352} {27} + \sqrt {\frac {17576} {729} + \frac {123904} {729} } }\)
\(\ds \) \(=\) \(\ds \sqrt [3] {\frac {352 + 376.138} {27} }\)
\(\ds \) \(=\) \(\ds 2.9988 \ldots\)
\(\ds T\) \(=\) \(\ds \sqrt [3] {R - \sqrt {Q^3 + R^2} }\)
\(\ds \) \(=\) \(\ds \sqrt [3] {\frac {352} {27} - \sqrt {\paren {\frac {26} 9}^3 + \paren {\frac {352} {27} }^2} }\)
\(\ds \) \(=\) \(\ds \sqrt [3] {\frac {352} {27} - \sqrt {\frac {17576} {729} + \frac {123904} {729} } }\)
\(\ds \) \(=\) \(\ds \sqrt [3] {\frac {352 - 376.138} {27} }\)
\(\ds \) \(=\) \(\ds -0.9633 \ldots\)
\(\ds x_1\) \(=\) \(\ds S + T - \dfrac b {3 a}\)
\(\ds \) \(=\) \(\ds 2.9988 \ldots - 0.9633 \ldots - \dfrac 2 3\)
\(\ds \) \(=\) \(\ds 1.3688 \ldots\)

The other two roots are complex.

$\blacksquare$


Sources

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