Question

Solution Sets of Quadratic Equations Quick Check

Answer 1

Certainly! Here’s a quick guide on how to determine the solution sets of quadratic equations.

### Quadratic Equation Format
A quadratic equation can be expressed in standard form as:
\[
ax^2 + bx + c = 0
\]
where \( a \), \( b \), and \( c \) are constants and \( a \neq 0 \).

### Methods to Solve Quadratic Equations

1. **Factoring**
- If the quadratic can be factored, write it as:
\[
(px + q)(rx + s) = 0
\]
- Set each factor to zero:
\[
px + q = 0 \quad \text{and} \quad rx + s = 0
\]
- Solve for \( x \).

2. **Quadratic Formula**
- If the quadratic is not easily factorable, use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
- Here, \( b^2 - 4ac \) is called the **discriminant**.

3. **Completing the Square**
- Rearrange the equation to isolate the quadratic and linear terms:
\[
ax^2 + bx = -c
\]
- Divide by \( a \):
\[
x^2 + \frac{b}{a}x = -\frac{c}{a}
\]
- Complete the square on the left-hand side and solve for \( x \).

### Analyzing the Discriminant
The discriminant \( D = b^2 - 4ac \) determines the nature of the roots:
- **If \( D > 0 \)**: Two distinct real solutions.
- **If \( D = 0 \)**: One real double solution (or two real solutions that are the same).
- **If \( D < 0 \)**: Two complex (imaginary) solutions.

### Example Problems

1. **Example 1**:
Solve \( x^2 - 5x + 6 = 0 \).
- Factor: \( (x - 2)(x - 3) = 0 \)
- Solutions: \( x = 2 \) and \( x = 3 \).

2. **Example 2**:
Solve \( 2x^2 + 4x + 2 = 0 \).
- Discriminant: \( D = 4^2 - 4\cdot2\cdot2 = 16 - 16 = 0 \).
- Solution: \( x = \frac{-4}{2\cdot2} = -1 \) (double root).

3. **Example 3**:
Solve \( x^2 + 2x + 5 = 0 \).
- Discriminant: \( D = 2^2 - 4\cdot1\cdot5 = 4 - 20 = -16 \).
- Solutions: \( x = \frac{-2 \pm \sqrt{-16}}{2\cdot1} = -1 \pm 2i \) (complex roots).

### Quick Check Questions
1. Solve \( x^2 - 4 = 0 \).
2. Solve \( 3x^2 + 12x + 12 = 0 \).
3. Determine the nature of the roots for \( x^2 + 6x + 10 = 0 \).

Feel free to ask for specific examples or further clarification on any of the methods!