Question - Solving a Cubic Equation with Fractions
Solution:
To solve the equation $$ x + \frac{1}{3x} = \frac{1}{x} - \frac{1}{3} $$, you can follow these steps:1. First, find a common denominator to combine the fractions on each side of the equation. The common denominator for the terms involving $$x$$ would be $$3x^2$$.2. Multiply each term on both sides by $$3x^2$$ to clear the fractions. This gives us:\[ 3x^3 + 1 = 3x - x^2 \]3. Rearrange the equation to form a cubic equation by moving all terms to one side:\[ 3x^3 + x^2 - 3x + 1 = 0 \]4. Now solve this cubic equation for $$x$$. Solving cubic equations analytically can be complex, involving Cardano's formula or other methods, but you can also solve it numerically or by factoring if possible.If you want to solve it by factoring, you'd look for values of $$x$$ that make the equation equal to zero. This can be done by trial and error (plugging in small integers for $$x$$ to see if they work). However, for this particular equation, factoring by inspection may be difficult, and it might be more efficient to use numerical methods or a graphing calculator to find the roots of the equation.If you find that the equation has a rational root, you could use the Rational Root Theorem to help find factors. The Rational Root Theorem states that any rational solution, when written in lowest terms $$p/q$$, must have $$p$$ as a factor of the constant term and $$q$$ as a factor of the leading coefficient. In this case, $$p$$ would be a factor of 1, and $$q$$ would be a factor of 3. Possible rational roots could be $$\pm1, \pm\frac{1}{3}$$.But for a precise solution without a calculator, you would need to apply complex methods beyond the scope of this explanation. If the equation were to have a simpler form, or if it were given that a certain method should be used, the approach might differ.
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