Let's identify which quadratic equation Ramiya's given formula corresponds to by systematically comparing it with each option.
### Step-by-Step Analysis:
1. Identify the coefficients:
- From her equation [tex]\( x = \frac{-3 \pm \sqrt{3^2 - 4(1)(2)}}{2(1)} \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 3 \)[/tex]
- [tex]\( c = 2 \)[/tex]
2. Calculate the discriminant:
- [tex]\( b^2 - 4ac \)[/tex]
- Here, [tex]\( b^2 - 4ac = 3^2 - 4(1)(2) = 9 - 8 = 1 \)[/tex].
Now, let's check each option's discriminant to see which one matches 1:
1. Equation 1: [tex]\( 0 = x^2 + 3x + 2 \)[/tex]
- Coefficients are: [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], [tex]\( c = 2 \)[/tex]
- Discriminant: [tex]\( b^2 - 4ac = 3^2 - 4(1)(2) = 9 - 8 = 1 \)[/tex]
2. Equation 2: [tex]\( 0 = x^2 - 3x + 2 \)[/tex]
- Coefficients are: [tex]\( a = 1 \)[/tex], [tex]\( b = -3 \)[/tex], [tex]\( c = 2 \)[/tex]
- Discriminant: [tex]\( b^2 - 4ac = (-3)^2 - 4(1)(2) = 9 - 8 = 1 \)[/tex]
3. Equation 3: [tex]\( 0 = 2x^2 + 3x + 1 \)[/tex]
- Coefficients are: [tex]\( a = 2 \)[/tex], [tex]\( b = 3 \)[/tex], [tex]\( c = 1 \)[/tex]
- Discriminant: [tex]\( b^2 - 4ac = 3^2 - 4(2)(1) = 9 - 8 = 1 \)[/tex]
4. Equation 4: [tex]\( 0 = 2x^2 - 3x + 1 \)[/tex]
- Coefficients are: [tex]\( a = 2 \)[/tex], [tex]\( b = -3 \)[/tex], [tex]\( c = 1 \)[/tex]
- Discriminant: [tex]\( b^2 - 4ac = (-3)^2 - 4(2)(1) = 9 - 8 = 1 \)[/tex]
### Conclusion:
All four equations provided have the same discriminant value of 1. However, Ramiya's equation specifically has [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 2 \)[/tex]. This matches the coefficients for the first equation, [tex]\( 0 = x^2 + 3x + 2 \)[/tex].
Thus, Ramiya's quadratic equation is [tex]\( 0 = x^2 + 3x + 2 \)[/tex].
