51.2=2 x x^3/2

asked by guest
on Oct 28, 2024 at 12:33 am



You asked:

Solve the equation \(51.2 = \frac{2 x \cdot {x}^{3}}{2}\) for the variable \(x\).

MathBot Answer:

The solutions are given by the formula: \[x = \frac{4 \cdot 5^{\frac{3}{4}} \cos{\left(\frac{\pi k}{2} \right)}}{5} + \frac{4 \cdot 5^{\frac{3}{4}} \sin{\left(\frac{\pi k}{2} \right)}}{5} i\] where \(k\) is an integer in \(0 \le k < 4\).

There are \(2\) real solutions.

There are \(2\) solutions with nonzero imaginary part.

Real solutions: \[\begin{aligned}x &= \frac{4 \cdot 5^{\frac{3}{4}}}{5} \\&\approx 2.6749612\\x &= - \frac{4 \cdot 5^{\frac{3}{4}}}{5} \\&\approx -2.6749612\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}x &= \frac{4 \cdot 5^{\frac{3}{4}}}{5} i \\&\approx 2.6749612 i\\x &= - \frac{4 \cdot 5^{\frac{3}{4}}}{5} i \\&\approx -2.6749612 i\end{aligned}\]

\(i\) is the imaginary unit, defined as \(i^2 = -1\).

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