If \(-nx^3 + tx + c = 0\), what is \(x\) equal to?
Answer (2)
0\ \ \Rightarrow\ \ \ x= \sqrt[3]{ \frac{c}{2n}- \sqrt\Delta} } +\sqrt[3]{ \frac{c}{2n}+ \sqrt\Delta} \\ \\\Delta=0\ \ \ \Rightarrow\ \ \ x_1=\sqrt[3]{ \frac{c}{2n}},\ \ \ \ x_2=-2\sqrt[3]{ \frac{c}{2n}}\\ \\\Delta<0\ \ \ \Rightarrow\ \ \ there\ is\ no\ solution\ for\ x\in R"> -nx^3 + tx + c = 0\ /:(-n)\ \ \ \wedge\ \ \ n \neq 0\\ \\x^3- \frac{t}{n} x-\frac{c}{n}=0\\ \\\Delta=(- \frac{t}{n})^3+(-\frac{c}{n})^2= \frac{-t^3}{n^3} + \frac{c^2}{n^2} = \frac{-t^3+n\cdot c^2}{n^3} \\ \\\Delta>0\ \ \Rightarrow\ \ \ x= \sqrt[3]{ \frac{c}{2n}- \sqrt\Delta} } +\sqrt[3]{ \frac{c}{2n}+ \sqrt\Delta} \\ \\\Delta=0\ \ \ \Rightarrow\ \ \ x_1=\sqrt[3]{ \frac{c}{2n}},\ \ \ \ x_2=-2\sqrt[3]{ \frac{c}{2n}}\\ \\\Delta<0\ \ \ \Rightarrow\ \ \ there\ is\ no\ solution\ for\ x\in R
To solve the equation − n x 3 + t x + c = 0 , rearrange it to find x as a function of the constants n, t, and c. The nature and number of solutions can be determined using the discriminant Δ . Depending on the sign of Δ , we can find either distinct real roots, a single repeated root, or conclude that other roots are complex.
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