Example Question - solving cubic equation
Here are examples of questions we've helped users solve.
Solving a Cubic Equation with a Given Value of x
The image shows two equations: 1) \( y = x^3 + 3x^2 - 24x + b \) 2) \( x_0 = 2 \) It appears that you have been provided with a cubic equation and a value for \( x_0 \), which typically would suggest either finding the value of y when \( x = x_0 \), or determining a constant in the equation, like 'b', based on some additional information about \( x_0 \). However, the context or the specific question you need to solve with these equations is not given. If the problem is to find the corresponding y-value (let's call it \( y_0 \)) when \( x = x_0 \), then we would substitute \( x = 2 \) into the first equation: \( y_0 = (2)^3 + 3(2)^2 - 24(2) + b \) \( y_0 = 8 + 3(4) - 48 + b \) \( y_0 = 8 + 12 - 48 + b \) \( y_0 = 20 - 48 + b \) \( y_0 = -28 + b \) So the \( y_0 \) value when \( x = 2 \) depends on the value of 'b'. However, if b is what you're supposed to find, then there is missing information. There needs to be additional information about the graph or a specific y-value when \( x = x_0 \) to find 'b'. If that's the case, please provide the additional information so I can assist further.
Solving a Cubic Equation with Fractions
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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