Math worksheet focusing on using the discriminant to determine the number of solutions for quadratic equations.

Problem Overview:

The task involves using the discriminant of a quadratic equation to determine the number of solutions. The general form of a quadratic equation is:

$$
ax^2 + bx + c = 0
$$

The discriminant, denoted by \( \Delta \), is given by:

$$
\Delta = b^2 - 4ac
$$

The number of solutions depends on the value of the discriminant:
- If \( \Delta > 0 \): The equation has 2 real solutions.
- If \( \Delta = 0 \): The equation has 1 real solution (a repeated root).
- If \( \Delta < 0 \): The equation has no real solutions (complex solutions).

Step-by-Step Solution:

We will solve each quadratic equation step by step, following the instructions provided in the image.

---

#### 1. \( 2x^2 + 5x + 3 = 0 \)

1. Identify coefficients:
- \( a = 2 \)
- \( b = 5 \)
- \( c = 3 \)

2. Write the discriminant formula:
$$
\Delta = b^2 - 4ac
$$

3. Substitute values into the formula:
$$
\Delta = (5)^2 - 4(2)(3)
$$

4. Simplify using the order of operations (PEMDAS):
$$
\Delta = 25 - 4(6) = 25 - 24 = 1
$$

5. Determine the number of solutions:
- Since \( \Delta = 1 > 0 \), there are 2 real solutions.

Answer:
$$
\boxed{2}
$$

---

#### 2. \( x^2 + 6x + 9 = 0 \)

1. Identify coefficients:
- \( a = 1 \)
- \( b = 6 \)
- \( c = 9 \)

2. Write the discriminant formula:
$$
\Delta = b^2 - 4ac
$$

3. Substitute values into the formula:
$$
\Delta = (6)^2 - 4(1)(9)
$$

4. Simplify using the order of operations (PEMDAS):
$$
\Delta = 36 - 4(9) = 36 - 36 = 0
$$

5. Determine the number of solutions:
- Since \( \Delta = 0 \), there is 1 real solution (a repeated root).

Answer:
$$
\boxed{1}
$$

---

#### 3. \( 4x^2 + 3x + 2 = 0 \)

1. Identify coefficients:
- \( a = 4 \)
- \( b = 3 \)
- \( c = 2 \)

2. Write the discriminant formula:
$$
\Delta = b^2 - 4ac
$$

3. Substitute values into the formula:
$$
\Delta = (3)^2 - 4(4)(2)
$$

4. Simplify using the order of operations (PEMDAS):
$$
\Delta = 9 - 4(8) = 9 - 32 = -23
$$

5. Determine the number of solutions:
- Since \( \Delta = -23 < 0 \), there are no real solutions.

Answer:
$$
\boxed{0}
$$

---

#### 4. \( 3x^2 + 6x + 3 = 0 \)

1. Identify coefficients:
- \( a = 3 \)
- \( b = 6 \)
- \( c = 3 \)

2. Write the discriminant formula:
$$
\Delta = b^2 - 4ac
$$

3. Substitute values into the formula:
$$
\Delta = (6)^2 - 4(3)(3)
$$

4. Simplify using the order of operations (PEMDAS):
$$
\Delta = 36 - 4(9) = 36 - 36 = 0
$$

5. Determine the number of solutions:
- Since \( \Delta = 0 \), there is 1 real solution (a repeated root).

Answer:
$$
\boxed{1}
$$

---

#### 5. \( 2x^2 - 4x + 2 = 0 \)

1. Identify coefficients:
- \( a = 2 \)
- \( b = -4 \)
- \( c = 2 \)

2. Write the discriminant formula:
$$
\Delta = b^2 - 4ac
$$

3. Substitute values into the formula:
$$
\Delta = (-4)^2 - 4(2)(2)
$$

4. Simplify using the order of operations (PEMDAS):
$$
\Delta = 16 - 4(4) = 16 - 16 = 0
$$

5. Determine the number of solutions:
- Since \( \Delta = 0 \), there is 1 real solution (a repeated root).

Answer:
$$
\boxed{1}
$$

---

#### 6. \( 3x^2 - 7x + 5 = 0 \)

1. Identify coefficients:
- \( a = 3 \)
- \( b = -7 \)
- \( c = 5 \)

2. Write the discriminant formula:
$$
\Delta = b^2 - 4ac
$$

3. Substitute values into the formula:
$$
\Delta = (-7)^2 - 4(3)(5)
$$

4. Simplify using the order of operations (PEMDAS):
$$
\Delta = 49 - 4(15) = 49 - 60 = -11
$$

5. Determine the number of solutions:
- Since \( \Delta = -11 < 0 \), there are no real solutions.

Answer:
$$
\boxed{0}
$$

---

#### 7. \( 2x^2 + 4x + 3 = 0 \)

1. Identify coefficients:
- \( a = 2 \)
- \( b = 4 \)
- \( c = 3 \)

2. Write the discriminant formula:
$$
\Delta = b^2 - 4ac
$$

3. Substitute values into the formula:
$$
\Delta = (4)^2 - 4(2)(3)
$$

4. Simplify using the order of operations (PEMDAS):
$$
\Delta = 16 - 4(6) = 16 - 24 = -8
$$

5. Determine the number of solutions:
- Since \( \Delta = -8 < 0 \), there are no real solutions.

Answer:
$$
\boxed{0}
$$

---

#### 8. \( 2x^2 - 4x + 3 = 0 \)

1. Identify coefficients:
- \( a = 2 \)
- \( b = -4 \)
- \( c = 3 \)

2. Write the discriminant formula:
$$
\Delta = b^2 - 4ac
$$

3. Substitute values into the formula:
$$
\Delta = (-4)^2 - 4(2)(3)
$$

4. Simplify using the order of operations (PEMDAS):
$$
\Delta = 16 - 4(6) = 16 - 24 = -8
$$

5. Determine the number of solutions:
- Since \( \Delta = -8 < 0 \), there are no real solutions.

Answer:
$$
\boxed{0}
$$

---

Final Answers:

1. \( 2x^2 + 5x + 3 = 0 \): \( \boxed{2} \)
2. \( x^2 + 6x + 9 = 0 \): \( \boxed{1} \)
3. \( 4x^2 + 3x + 2 = 0 \): \( \boxed{0} \)
4. \( 3x^2 + 6x + 3 = 0 \): \( \boxed{1} \)
5. \( 2x^2 - 4x + 2 = 0 \): \( \boxed{1} \)
6. \( 3x^2 - 7x + 5 = 0 \): \( \boxed{0} \)
7. \( 2x^2 + 4x + 3 = 0 \): \( \boxed{0} \)
8. \( 2x^2 - 4x + 3 = 0 \): \( \boxed{0} \)