(1+x)^4=1.04

asked by guest
on Nov 23, 2024 at 11:45 pm



You asked:

Solve the equation \({\left( 1 + x \right)}^{4} = 1.04\) for the variable \(x\).

MathBot Answer:

The solutions are given by the formula: \[x = \left(-1 + \frac{\sqrt{5} \cdot \sqrt[4]{26} \cos{\left(\frac{\pi k}{2} \right)}}{5}\right) + \frac{\sqrt{5} \cdot \sqrt[4]{26} \sin{\left(\frac{\pi k}{2} \right)}}{5} i\] 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: \[\begin{aligned}x &= -1 + \frac{\sqrt{5} \cdot \sqrt[4]{26}}{5} \\&\approx 0.0098534065\\x &= -1 - \frac{\sqrt{5} \cdot \sqrt[4]{26}}{5} \\&\approx -2.0098534\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}x &= -1 + \frac{\sqrt[4]{26} \sqrt{5} i}{5} \\&= -1 + 1.0098534 i\\x &= -1 - \frac{\sqrt[4]{26} \sqrt{5} i}{5} \\&= -1 -1.0098534 i\end{aligned}\]

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

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