Solve the equation after making an appropriate substitution. Complete parts a through d.

4 x Superscript two thirds Baseline minus 13 x Superscript one third Baseline plus 3 equals 04x23−13x13+3=0

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Part 1

​a) Determine the appropriate substitution using the new variable u.

uequals=x Superscript one thirdx13

Part 2

​b) Write the new quadratic equation using the variable.

4 u squared minus 13 u plus 34u2−13u+3equals=0

​(Simplify your answer. Do not​ factor.)

Part 3

​c) Solve the new equation for the variable u.

uequals=enter your response here

​(Simplify your answer. Type your​ answer(s) as integers or simplified fractions. Use a comma to separate answers as​ needed.)

### 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 \( x^{\frac{2}{3}} = (x^{\frac{1}{3}})^2 = u^2 \). 

---

### 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 \( u \).

---

### 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:
\[
u = 3 \quad \text{or} \quad u = \frac{1}{4}
\]

---

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!

Solve the equation after making an appropriate substitution. Complete parts a through d.

4x^(2/3) − 13x^(1/3) + 3 = 0

Part 1
a) Determine the appropriate substitution using the new variable u.
u = x^(1/3)

Part 2
b) Write the new quadratic equation using the variable u.
4u^2 − 13u + 3 = 0

Part 3
c) Solve the new equation for the variable u.

Learn how to solve the equation 4x^(2/3) − 13x^(1/3) + 3 using substitution and the quadratic formula. This step-by-step solution covers key algebraic concepts such as substitution and quadratic equations, suitable for Grades 9-10.

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