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The problem asks for the truth value of \( \forall y Q(y) \), where \( Q(y) \) is defined as the inequality \( y^2 - 9y + 25 > 0 \), and \( U \) is the set of real numbers. To answer this, let's go step by step.

### Step 1: Analyze \( Q(y) \)
The expression given for \( Q(y) \) is \( y^2 - 9y + 25 > 0 \). This is a quadratic inequality. To determine where this inequality holds true, let's first find the roots of the quadratic equation \( y^2 - 9y + 25 = 0 \).

#### Step 1.1: Solve the quadratic equation
The general form of a quadratic equation is \( ax^2 + bx + c = 0 \). In this case, \( a = 1 \), \( b = -9 \), and \( c = 25 \).

Using the quadratic formula:
\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Substitute the values of \( a \), \( b \), and \( c \):
\[
y = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(25)}}{2(1)}
\]
\[
y = \frac{9 \pm \sqrt{81 - 100}}{2}
\]
\[
y = \frac{9 \pm \sqrt{-19}}{2}
\]
Since the discriminant (\( \Delta = 81 - 100 = -19 \)) is negative, the quadratic equation has no real roots. This means the parabola represented by \( y^2 - 9y + 25 \) does not intersect the x-axis.

### Step 2: Nature of the quadratic expression
Because the quadratic equation has no real roots and the leading coefficient of \( y^2 \) is positive (indicating the parabola opens upwards), the quadratic expression \( y^2 - 9y + 25 \) is **always positive** for all real values of \( y \).

### Step 3: Truth value of \( \forall y Q(y) \)
Since \( y^2 - 9y + 25 > 0 \) holds true for all real values of \( y \), the truth value of \( \forall y Q(y) \) is **true**.

### Conclusion
The inequality \( y^2 - 9y + 25 > 0 \) is always satisfied for all real numbers \( y \). Therefore, the truth value of \( \forall y Q(y) \) is **true**.

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Would you like further details on quadratic inequalities or another part of the solution explained in more depth?

Here are 5 related questions to expand on this topic:

1. How does the discriminant determine the nature of a quadratic equation's roots?
2. What is the geometric interpretation of a quadratic function with no real roots?
3. How do we solve quadratic inequalities in general?
4. Can you provide examples of when \( Q(y) \) would not hold true for all real numbers?
5. How does the concept of universal quantification apply to this problem?

**Tip**: For solving quadratic inequalities, graphing the function can help visualize where the inequality holds.

Let Q(y) = {y^2 - 9y + 25 > 0} and U is the set of real numbers. What is the truth value of ∀y Q(y)?

Explore the truth value of the inequality y^2 - 9y + 25 > 0 for all real numbers. This problem uses the quadratic formula and analyzes the nature of the discriminant to determine the truth value. Ideal for students in Grades 9-12.

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