Questions: Find the real solutions of the equation. 7 x^(2 / 3) - 48 x^(1 / 3) - 7 = 0 What is the solution set? Select the correct choice below and fill in any answer boxes within A. (Use a comma to separate answers as needed. Type an integer or a simplified fraction) B. There are no solutions.

Solution Steps

To solve the equation \(7 x^{2/3} - 48 x^{1/3} - 7 = 0\), we can use a substitution method. Let \(y = x^{1/3}\). Then the equation becomes a quadratic equation in terms of \(y\): \(7y^2 - 48y - 7 = 0\). We can solve this quadratic equation using the quadratic formula. Once we find the values of \(y\), we can substitute back to find the corresponding values of \(x\).

To solve the equation \(7x^{2/3} - 48x^{1/3} - 7 = 0\), we first make a substitution. Let \(y = x^{1/3}\). This transforms the equation into a quadratic form: \[ 7y^2 - 48y - 7 = 0 \]

We solve the quadratic equation \(7y^2 - 48y - 7 = 0\) using the quadratic formula: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 7\), \(b = -48\), and \(c = -7\). The solutions for \(y\) are: \[ y_1 = -0.1429 \] \[ y_2 = 7.0000 \]

Since \(y = x^{1/3}\), we find \(x\) by cubing the solutions for \(y\): \[ x_1 = (-0.1429)^3 = -0.0029 \] \[ x_2 = (7.0000)^3 = 343.0000 \]

Final Answer