Questions: Solve the following equation by making an appropriate substitution. 6x^(2/3) - 5x^(1/3) - 1 = 0 Solve the equation. Select the correct choice below and, if necessary, fill in the answer box to complete your choice A. The solution set is . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The solution set is the empty set.

Solve the following equation by making an appropriate substitution. 6x^(2/3) - 5x^(1/3) - 1 = 0 Solve the equation. Select the correct choice below and, if necessary, fill in the answer box to complete your choice A. The solution set is . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The solution set is the empty set.
Transcript text: Solve the following equation by making an appropriate substitution. \[ 6 x^{\frac{2}{3}}-5 x^{\frac{1}{3}}-1=0 \] Solve the equation. Select the correct choice below and, if necessary, fill in the answer box to complete your choice A. The solution set is $\square$ \}. (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The solution set is the empty set.
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Solution

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Solution Steps

Step 1: Make an appropriate substitution

Given the equation \(ax^{\frac{2}{3}} + bx^{\frac{1}{3}} + c = 0\), we substitute \(u = x^{\frac{1}{3}}\), transforming the equation into \(au^2 + bu + c = 0\).

Step 2: Solve the quadratic equation

We solve the quadratic equation, finding \(u_1 = 1\) and \(u_2 = -0.167\).

Step 3: Back-substitute to find the values of \(x\)

Back-substituting, we find \(x_1 = u_1^3 = 1\) and \(x_2 = u_2^3 = -0.00463\).

Final Answer: \(x_1 = 1\), \(x_2 = -0.00463\)

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