Part 1: Substitution

We are given the equation:
$4x^{\frac{2}{3}} - 13x^{\frac{1}{3}} + 3 = 0$

The appropriate substitution is: $u = x^{\frac{1}{3}}$

This is because $x^{\frac{2}{3}} = (x^{\frac{1}{3}})^2 = u^2$.

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Part 2: Writing the New Quadratic Equation

Substitute $u = x^{\frac{1}{3}}$ into the original equation: $4x^{\frac{2}{3}} - 13x^{\frac{1}{3}} + 3 = 0$

This becomes: $4u^2 - 13u + 3 = 0$

This is the new quadratic equation in terms of $u$.

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Part 3: Solving the Quadratic Equation

We now solve the quadratic equation: $4u^2 - 13u + 3 = 0$

We can use the quadratic formula to solve for $u$: $u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Where $a = 4$, $b = -13$, and $c = 3$.

First, calculate the discriminant: $b^2 - 4ac = (-13)^2 - 4(4)(3) = 169 - 48 = 121$

Now, apply the quadratic formula: $u = \frac{-(-13) \pm \sqrt{121}}{2(4)} = \frac{13 \pm 11}{8}$

This gives us two solutions for $u$: $u_1 = \frac{13 + 11}{8} = \frac{24}{8} = 3$ $u_2 = \frac{13 - 11}{8} = \frac{2}{8} = \frac{1}{4}$

Thus, the solutions for $u$ are: $u = 3 \quad \text{or} \quad u = \frac{1}{4}$

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Would you like help with the remaining part or further details? Here are some related questions:

1. How do you reverse the substitution to find $x$?
2. How do we verify the solutions for $u$?
3. Can you use factoring instead of the quadratic formula here?
4. What is the discriminant's role in determining the type of solutions?
5. How does the form of the original equation relate to solving it by substitution?

Tip: Always check your discriminant value first in the quadratic formula—it helps to predict if you’ll get real or complex solutions!