1.02^2=(1+(x/4))^4

asked by guest
on Jan 15, 2025 at 4:30 am



You asked:

Solve the equation \({1.02}^{2} = {\left( 1 + \frac{x}{4} \right)}^{4}\) for the variable \(x\).

MathBot Answer:

The solutions are given by the formula: \[x = \left(-4 + \frac{2 \sqrt{102} \cos{\left(\frac{\pi k}{2} \right)}}{5}\right) + \frac{2 \sqrt{102} \sin{\left(\frac{\pi k}{2} \right)}}{5} i\] 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: \[\begin{aligned}x &= -4 + \frac{2 \sqrt{102}}{5} \\&\approx 0.039801975\\x &= -4 - \frac{2 \sqrt{102}}{5} \\&\approx -8.039802\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}x &= -4 + \frac{2 \sqrt{102} i}{5} \\&= -4 + 4.039802 i\\x &= -4 - \frac{2 \sqrt{102} i}{5} \\&= -4 -4.039802 i\end{aligned}\]

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

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