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Let's analyze the problem and find the truth values for each of the given quantified statements.

We are given the open statements:
- \( p(x): x^2 = 2x \)
- \( q(x): x^3 = 4x \)
with the set of all integers as the universe.

### Step 1: Analyze \( p(x) \) and \( q(x) \) in terms of integers
1. \( p(x): x^2 = 2x \)
   - Factor: \( x^2 - 2x = 0 \Rightarrow x(x - 2) = 0 \)
   - Solutions: \( x = 0 \) or \( x = 2 \)

2. \( q(x): x^3 = 4x \)
   - Factor: \( x^3 - 4x = 0 \Rightarrow x(x^2 - 4) = 0 \Rightarrow x(x - 2)(x + 2) = 0 \)
   - Solutions: \( x = 0, 2, -2 \)

### Step 2: Evaluate each statement

#### (i) \( \forall x, p(x) \)  
This means that \( p(x) \) is true for all integers \( x \). But \( p(x) \) is only true for \( x = 0 \) and \( x = 2 \), so it is **false** for all other integers.

**Truth value: False**

#### (ii) \( \exists x, q(x) \)  
This means there exists an \( x \) such that \( q(x) \) is true. \( q(x) \) is true for \( x = 0, 2, -2 \), so this statement is **true**.

**Truth value: True**

#### (iii) \( \forall x, \neg p(x) \)  
This means that \( p(x) \) is false for all \( x \). But \( p(x) \) is true for \( x = 0 \) and \( x = 2 \), so this statement is **false**.

**Truth value: False**

#### (iv) \( \exists x, \neg q(x) \)  
This means there exists an \( x \) such that \( q(x) \) is false. \( q(x) \) is true only for \( x = 0, 2, -2 \), so for all other integers \( q(x) \) is false. Hence, this statement is **true**.

**Truth value: True**

#### (v) \( \forall x, p(x) \land q(x) \)  
This means that for all \( x \), both \( p(x) \) and \( q(x) \) are true. The only \( x \) that satisfies both \( p(x) \) and \( q(x) \) is \( x = 2 \). Thus, for all other integers, the conjunction is false, so this statement is **false**.

**Truth value: False**

#### (vi) \( \exists x, p(x) \land q(x) \)  
This means there exists an \( x \) such that both \( p(x) \) and \( q(x) \) are true. For \( x = 2 \), both \( p(x) \) and \( q(x) \) are true, so this statement is **true**.

**Truth value: True**

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### Final Truth Values:
(i) False  
(ii) True  
(iii) False  
(iv) True  
(v) False  
(vi) True  

Let me know if you need any details or further explanations! 

Here are 5 related questions to expand your understanding:
1. How do you determine if a statement is true for all integers?
2. What happens when we change the domain to real numbers for these statements?
3. How does negation affect the truth value of quantified statements?
4. Can a different factoring approach affect the solution sets of \( p(x) \) or \( q(x) \)?
5. How would these truth values change if the set of integers is limited (e.g., to non-negative integers)?

**Tip:** When working with quantifiers, always focus on the exact meaning of the logical statement—whether it's referring to all possible values or just some values within a specified set.

Let p(x) be the open statement 'x^2 = 2x' and q(x) be the open statement 'x^3 = 4x' with the set of all integers as the universe. Write down the truth values of the following quantified statements: (i) ∀x, p(x) (ii) ∃x, q(x) (iii) ∀x, ¬p(x) (iv) ∃x, ¬q(x) (v) ∀x, p(x) ∧ q(x) (vi) ∃x, p(x) ∧ q(x).

Determine the truth values for quantified statements involving p(x): x^2 = 2x and q(x): x^3 = 4x over the set of integers. Includes evaluation of universal and existential quantifiers for mathematical logic and factoring problems. Suitable for high school students in Grades 10-12.

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