https://r2.erweima.ai/mathbot/image/1725560271078i587jh4g.jpg

I have opened the image you provided. I'll now extract the mathematical problems to help you solve them.

Here are the questions from the image:

1. Factorize: \( 12x^2 - 6xy + y^2 - 9y^2 + 2x + 5 \)
2. A person has walked 360 km in 16 days at a rate of five hours per day. How many days will it take to walk 690 km, walking 7 hours per day?
3. If \( m = t^2 \), \( n = t^3 + 1 \), and \( z = t^2 + 1 \), calculate \( A = (mn)^5 \).
4. Using logical equivalences, solve \( (p \rightarrow q) \land (q \rightarrow r) \land (r \rightarrow s) \leftrightarrow (p \rightarrow s) \land (p \rightarrow r) \land (p \rightarrow n) \).
5. Factorize: \( 3x^2 - 5x + 2 = 0 \).

I will begin solving them one by one. Let’s start with question 1 (factorization).The expression \( 12x^2 - 6xy + y^2 - 9y^2 + 2x + 5 \) simplifies to \( 12x^2 - 6xy + 2x - 8y^2 + 5 \). This expression is already in its simplest factored form and cannot be further factorized meaningfully over the real numbers.

Next, let's solve question 2 about calculating the number of days it will take to walk 690 km, given the details provided.It will take approximately 21.9 days for the person to walk 690 km at a pace of 7 hours per day.

Next, I will solve question 3, where we need to calculate \( A = (mn)^5 \) given the expressions \( m = t^2 \), \( n = t^3 + 1 \), and \( z = t^2 + 1 \).It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to continue solving the rest of the problems step by step manually, or perhaps you'd like to focus on a specific question?

In the meantime, here are some relative questions you might find interesting:
1. Can you clarify how to calculate large powers like \( (mn)^5 \) efficiently?
2. What is the concept of logical equivalences in the fourth question?
3. How do we approach factorization for more complex quadratics like in question 5?
4. What is the significance of time in distance and speed problems?
5. How can we further simplify expressions for different variables?

**Tip:** When working with large powers like \( (mn)^5 \), expanding the expressions step by step can help reduce complexity and avoid mistakes.

1. Factorize: 12x^2 - 6xy + y^2 - 9y^2 + 2x + 5. 2. A person has walked 360 km in 16 days at a rate of five hours per day. How many days will it take to walk 690 km, walking 7 hours per day? 3. If m = t^2, n = t^3 + 1, and z = t^2 + 1, calculate A = (mn)^5. 4. Using logical equivalences, solve (p → q) ∧ (q → r) ∧ (r → s) ↔ (p → s) ∧ (p → r) ∧ (p → n). 5. Factorize: 3x^2 - 5x + 2 = 0.

Solve factorization problems, distance calculations, algebraic powers, and propositional logic using logical equivalences. This guide is designed for students in grades 9-12, covering key concepts and formulas.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation