To determine which equation has the coefficients [tex]\( a = -2 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = 3 \)[/tex], we need to carefully examine each of the given equations and compare them to the general form of a quadratic equation:
[tex]\[ 0 = ax^2 + bx + c \][/tex]
Let's analyze each option one by one:
1. Equation: [tex]\( 0 = -2x^2 + x + 3 \)[/tex]
- Comparing this with the standard form [tex]\( 0 = ax^2 + bx + c \)[/tex]:
- [tex]\( a = -2 \)[/tex]
- [tex]\( b = 1 \)[/tex]
- [tex]\( c = 3 \)[/tex]
This matches the given coefficients exactly: [tex]\( a = -2 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = 3 \)[/tex].
2. Equation: [tex]\( 0 = 2x^2 + x + 3 \)[/tex]
- Comparing this with the standard form [tex]\( 0 = ax^2 + bx + c \)[/tex]:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 1 \)[/tex]
- [tex]\( c = 3 \)[/tex]
Here, [tex]\( a \neq -2 \)[/tex]. So, this does not match the given coefficients.
3. Equation: [tex]\( 0 = -2x^2 + 3 \)[/tex]
- Comparing this with the standard form [tex]\( 0 = ax^2 + bx + c \)[/tex]:
- [tex]\( a = -2 \)[/tex]
- The term with [tex]\( x \)[/tex] is missing, so [tex]\( b = 0 \)[/tex]
- [tex]\( c = 3 \)[/tex]
Here, [tex]\( b \neq 1 \)[/tex]. So, this does not match the given coefficients.
4. Equation: [tex]\( 0 = 2x^2 - x + 3 \)[/tex]
- Comparing this with the standard form [tex]\( 0 = ax^2 + bx + c \)[/tex]:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = -1 \)[/tex]
- [tex]\( c = 3 \)[/tex]
Here, neither [tex]\( a \neq -2 \)[/tex] nor [tex]\( b \neq 1 \)[/tex]. So, this does not match the given coefficients.
Upon reviewing all the options, the equation that has an [tex]\( a \)[/tex]-value of -2, a [tex]\( b \)[/tex]-value of 1, and a [tex]\( c \)[/tex]-value of 3 is:
[tex]\[ 0 = -2x^2 + x + 3 \][/tex]
Therefore, the correct equation is:
[tex]\[ 0 = -2x^2 + x + 3 \][/tex]
