https://file.aiquickdraw.com/mathbot/file/1725471498028gcxulhb1.jpg

I see that you've uploaded an image containing math problems. I'll extract the content and solve each question one by one.

### Problem Breakdown:

The image contains the following problems:
1. **Determine the nature of roots of the quadratic equation \(5x^2 - 20x - 27 = 0\):**
   - A. Imaginary and unequal
   - B. Real and unequal
   - C. Real and equal
   - D. None of these

2. **Find the value of \( x \) in \( x^2 = 16 \):**
   - A. 4
   - B. \( \pm 4 \)
   - C. 8
   - D. \( \pm 8 \)

3. **Solve the equation \( x^2 + 6x - 7 = 0 \):**
   - A. 1, -7
   - B. -7, -1
   - C. -1, 7
   - D. 7, 1

4. **Solve the inequality \( 5(x + 3) > 4(x - 1) \):**
   - A. \( x > -7 \)
   - B. \( x < 7 \)
   - C. \( x < -7 \)
   - D. \( x > 7 \)

5. **Solve the inequality \( 9(x + 1) \geq 7(2x + 3) \):**
   - A. \( x \geq -2 \)
   - B. \( x \leq 2 \)
   - C. \( x \geq 2 \)
   - D. \( x \leq -2 \)

6. **Solve the inequality \( 6(x + 3) \leq 5(2x + 4) \):**
   - A. \( x \geq -7 \)
   - B. \( x \geq -9 \)
   - C. \( x \leq -7 \)
   - D. \( x \leq -9 \)

I'll now proceed to solve each problem systematically.Here are the solutions to each problem:

1. **Determine the nature of roots of the quadratic equation \(5x^2 - 20x - 27 = 0\):**
   - **Discriminant = 940.**
   - Since the discriminant is positive, the roots are **real and unequal**.
   - **Answer: B. Real and unequal**

2. **Find the value of \( x \) in \( x^2 = 16 \):**
   - The solutions are \( x = -4 \) and \( x = 4 \).
   - **Answer: B. \( \pm 4 \)**

3. **Solve the equation \( x^2 + 6x - 7 = 0 \):**
   - The solutions are \( x = -7 \) and \( x = 1 \).
   - **Answer: A. -7, 1**

4. **Solve the inequality \( 5(x + 3) > 4(x - 1) \):**
   - The solution is \( x > -19 \).
   - **Answer: Not listed directly, but implies \( x > -19 \)**.

5. **Solve the inequality \( 9(x + 1) \geq 7(2x + 3) \):**
   - The solution is \( x \leq -\frac{12}{5} \) or \( x \leq -2.4 \).
   - **Answer: D. \( x \leq -2 \)** (Closest option)

6. **Solve the inequality \( 6(x + 3) \leq 5(2x + 4) \):**
   - The solution is \( x \geq -\frac{1}{2} \).
   - **Answer: A. \( x \geq -7 \)** (Closest option, depending on how it is interpreted)

If you need further details or clarifications, let me know!

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### Relative Questions:
1. How do you determine the discriminant of a quadratic equation?
2. What does the sign of the discriminant tell us about the roots of a quadratic equation?
3. What methods can be used to solve quadratic equations aside from factoring?
4. How do you solve linear inequalities, and what rules must be followed when manipulating them?
5. What are some real-world applications of solving inequalities?

### Tip:
Always check if the quadratic equation can be factored easily before applying the quadratic formula. It can save time!

A set of math problems involving quadratic equations, inequalities, and the nature of roots.

Step-by-step solutions for solving quadratic equations, determining the nature of roots, and working with inequalities. This practice covers key algebra concepts for students in Grades 8-10.

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