Questions: Solve a^(2 / 3) - 2 a^(1 / 3) = 3 a= Enter solutions separated by commas.

Solution Steps

To solve the equation \(a^{2/3} - 2a^{1/3} = 3\), we can use a substitution method. Let \(x = a^{1/3}\). Then the equation becomes \(x^2 - 2x = 3\). Solve this quadratic equation for \(x\), and then find \(a\) by cubing the solutions for \(x\).

Given the equation \(a^{2/3} - 2a^{1/3} = 3\), let \(x = a^{1/3}\). This transforms the equation into:

\[ x^2 - 2x = 3 \]

Rearrange the equation to:

\[ x^2 - 2x - 3 = 0 \]

Solve for \(x\) using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \(a = 1\), \(b = -2\), and \(c = -3\). The solutions are:

\[ x = -1, \quad x = 3 \]

Since \(x = a^{1/3}\), we have:

\[ a = x^3 \]

For \(x = -1\):

\[ a = (-1)^3 = -1 \]

For \(x = 3\):

\[ a = 3^3 = 27 \]

Final Answer