solve a*x^4 + b*x = c for x

asked by guest
on Jan 23, 2025 at 2:37 am



You asked:

Solve the equation ax4+bx=ca {x}^{4} + b x = c for the unknown xx.

MathBot Answer:

x={−−−b2a232−−b2a23+2ba−−b2a232for ca=0−2c327a3+b4256a4+b216a23−2c3ac327a3+b4256a4+b216a232−−2c327a3+b4256a4+b216a23+2ba2c327a3+b4256a4+b216a23−2c3ac327a3+b4256a4+b216a23+2c3ac327a3+b4256a4+b216a232otherwisex={−−−b2a232+−b2a23+2ba−−b2a232for ca=0−2c327a3+b4256a4+b216a23−2c3ac327a3+b4256a4+b216a232+−2c327a3+b4256a4+b216a23+2ba2c327a3+b4256a4+b216a23−2c3ac327a3+b4256a4+b216a23+2c3ac327a3+b4256a4+b216a232otherwisex={−−b2a232−−b2a23−2ba−−b2a232for ca=02c327a3+b4256a4+b216a23−2c3ac327a3+b4256a4+b216a232−−2c327a3+b4256a4+b216a23−2ba2c327a3+b4256a4+b216a23−2c3ac327a3+b4256a4+b216a23+2c3ac327a3+b4256a4+b216a232otherwisex={−−b2a232+−b2a23−2ba−−b2a232for ca=02c327a3+b4256a4+b216a23−2c3ac327a3+b4256a4+b216a232+−2c327a3+b4256a4+b216a23−2ba2c327a3+b4256a4+b216a23−2c3ac327a3+b4256a4+b216a23+2c3ac327a3+b4256a4+b216a232otherwise\begin{aligned}x &= \begin{cases} - \frac{\sqrt{- \sqrt[3]{- \frac{b^{2}}{a^{2}}}}}{2} - \frac{\sqrt{\sqrt[3]{- \frac{b^{2}}{a^{2}}} + \frac{2 b}{a \sqrt{- \sqrt[3]{- \frac{b^{2}}{a^{2}}}}}}}{2} & \text{for}\: \frac{c}{a} = 0 \\- \frac{\sqrt{2 \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}} - \frac{2 c}{3 a \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}} + \frac{2 b}{a \sqrt{2 \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}} - \frac{2 c}{3 a \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}}}}} + \frac{2 c}{3 a \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}}}}}{2} & \text{otherwise} \end{cases}\\x &= \begin{cases} - \frac{\sqrt{- \sqrt[3]{- \frac{b^{2}}{a^{2}}}}}{2} + \frac{\sqrt{\sqrt[3]{- \frac{b^{2}}{a^{2}}} + \frac{2 b}{a \sqrt{- \sqrt[3]{- \frac{b^{2}}{a^{2}}}}}}}{2} & \text{for}\: \frac{c}{a} = 0 \\- \frac{\sqrt{2 \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}} - \frac{2 c}{3 a \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}} + \frac{2 b}{a \sqrt{2 \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}} - \frac{2 c}{3 a \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}}}}} + \frac{2 c}{3 a \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}}}}}{2} & \text{otherwise} \end{cases}\\x &= \begin{cases} \frac{\sqrt{- \sqrt[3]{- \frac{b^{2}}{a^{2}}}}}{2} - \frac{\sqrt{\sqrt[3]{- \frac{b^{2}}{a^{2}}} - \frac{2 b}{a \sqrt{- \sqrt[3]{- \frac{b^{2}}{a^{2}}}}}}}{2} & \text{for}\: \frac{c}{a} = 0 \\\frac{\sqrt{2 \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}} - \frac{2 c}{3 a \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}} - \frac{2 b}{a \sqrt{2 \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}} - \frac{2 c}{3 a \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}}}}} + \frac{2 c}{3 a \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}}}}}{2} & \text{otherwise} \end{cases}\\x &= \begin{cases} \frac{\sqrt{- \sqrt[3]{- \frac{b^{2}}{a^{2}}}}}{2} + \frac{\sqrt{\sqrt[3]{- \frac{b^{2}}{a^{2}}} - \frac{2 b}{a \sqrt{- \sqrt[3]{- \frac{b^{2}}{a^{2}}}}}}}{2} & \text{for}\: \frac{c}{a} = 0 \\\frac{\sqrt{2 \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}} - \frac{2 c}{3 a \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}} - \frac{2 b}{a \sqrt{2 \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}} - \frac{2 c}{3 a \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}}}}} + \frac{2 c}{3 a \sqrt[3]{\sqrt{\frac{c^{3}}{27 a^{3}} + \frac{b^{4}}{256 a^{4}}} + \frac{b^{2}}{16 a^{2}}}}}}{2} & \text{otherwise} \end{cases}\end{aligned}

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