6.75=(0.5-3a)^3/(0.4+a)(0.1+2a)^2

asked by guest
on Oct 28, 2024 at 12:26 pm



You asked:

Solve the equation \(6.75 = \frac{{\left( 0.5 - 3 a \right)}^{3}}{\left(0.4 + a\right) \cdot {\left( 0.1 + 2 a \right)}^{2}}\) for the variable \(a\).

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}a &= \sqrt[3]{\frac{49}{54000} + \frac{13 \sqrt{44382}}{864000}} - \frac{151}{7200 \sqrt[3]{\frac{49}{54000} + \frac{13 \sqrt{44382}}{864000}}} \approx 0.02847767\\a &= - \frac{\sqrt[3]{\frac{49}{54000} + \frac{13 \sqrt{44382}}{864000}}}{2} + \frac{151}{14400 \sqrt[3]{\frac{49}{54000} + \frac{13 \sqrt{44382}}{864000}}} + i \left(\frac{151 \sqrt{3}}{14400 \sqrt[3]{\frac{49}{54000} + \frac{13 \sqrt{44382}}{864000}}} + \frac{\sqrt{3} \sqrt[3]{\frac{49}{54000} + \frac{13 \sqrt{44382}}{864000}}}{2}\right) \approx -0.014238835 + 0.25204146 i\\a &= - \frac{\sqrt[3]{\frac{49}{54000} + \frac{13 \sqrt{44382}}{864000}}}{2} + \frac{151}{14400 \sqrt[3]{\frac{49}{54000} + \frac{13 \sqrt{44382}}{864000}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{49}{54000} + \frac{13 \sqrt{44382}}{864000}}}{2} - \frac{151 \sqrt{3}}{14400 \sqrt[3]{\frac{49}{54000} + \frac{13 \sqrt{44382}}{864000}}}\right) \approx -0.014238835 -0.25204146 i\end{aligned}\]

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

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