Alright, let's take a step-by-step approach to solving the problem of determining which equations have a leading coefficient of 3 and a constant term of -2. Here are the given equations:
1. [tex]\( 0 = 3x^2 + 2x - 2 \)[/tex]
2. [tex]\( 0 = -2 - 3x^2 + 3 \)[/tex]
3. [tex]\( 0 = -3x + 3x^2 - 2 \)[/tex]
4. [tex]\( 0 = 3x^2 + x + 2 \)[/tex]
5. [tex]\( 0 = -x - 2 + 3x^2 \)[/tex]
Step 1: Identify the leading coefficient and constant term of each equation.
1. For [tex]\( 0 = 3x^2 + 2x - 2 \)[/tex]:
- Leading coefficient (coefficient of [tex]\( x^2 \)[/tex]) is 3.
- Constant term is -2.
2. For [tex]\( 0 = -2 - 3x^2 + 3 \)[/tex]:
- After rearranging: [tex]\( 0 = 1 - 3x^2 \)[/tex]
- Leading coefficient is -3.
- Constant term is 1.
3. For [tex]\( 0 = -3x + 3x^2 - 2 \)[/tex]:
- Leading coefficient is 3.
- Constant term is -2.
4. For [tex]\( 0 = 3x^2 + x + 2 \)[/tex]:
- Leading coefficient is 3.
- Constant term is 2.
5. For [tex]\( 0 = -x - 2 + 3x^2 \)[/tex]:
- After rearranging: [tex]\( 0 = 3x^2 - x - 2 \)[/tex]
- Leading coefficient is 3.
- Constant term is -2.
Step 2: Check which equations meet the criteria (leading coefficient = 3 and constant term = -2).
- Equation 1: [tex]\( 0 = 3x^2 + 2x - 2 \)[/tex]
- Leading coefficient = 3
- Constant term = -2
- Matches the criteria.
- Equation 2: [tex]\( 0 = 1 - 3x^2 \)[/tex]
- Leading coefficient = -3
- Constant term = 1
- Does not match the criteria.
- Equation 3: [tex]\( 0 = -3x + 3x^2 - 2 \)[/tex]
- Leading coefficient = 3
- Constant term = -2
- Matches the criteria.
- Equation 4: [tex]\( 0 = 3x^2 + x + 2 \)[/tex]
- Leading coefficient = 3
- Constant term = 2
- Does not match the criteria.
- Equation 5: [tex]\( 0 = 3x^2 - x - 2 \)[/tex]
- Leading coefficient = 3
- Constant term = -2
- Matches the criteria.
Conclusion:
The equations that have a leading coefficient of 3 and a constant term of -2 are:
- [tex]\( 0 = 3x^2 + 2x - 2 \)[/tex]
- [tex]\( 0 = -3x + 3x^2 - 2 \)[/tex]
- [tex]\( 0 = 3x^2 - x - 2 \)[/tex]
These are the correct equations that meet the given criteria.
