To solve this problem, let's identify the standard form of a quadratic equation, which is:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
We are given the coefficients:
[tex]\[ a = 4, \][/tex]
[tex]\[ b = 5, \][/tex]
[tex]\[ c = -2. \][/tex]
We need to match these coefficients to one of the provided equations.
Let's examine each equation:
1. [tex]\( 4x^2 - 2x + 7 = 0 \)[/tex]
- Here, the coefficients are:
- [tex]\( a = 4, \)[/tex]
- [tex]\( b = -2, \)[/tex]
- [tex]\( c = 7. \)[/tex]
- These do not match with [tex]\( a = 4, b = 5, c = -2 \)[/tex].
2. [tex]\( x^2 + 2x = 0 \)[/tex]
- Here, the coefficients are:
- [tex]\( a = 1, \)[/tex]
- [tex]\( b = 2, \)[/tex]
- [tex]\( c = 0. \)[/tex]
- These do not match with [tex]\( a = 4, b = 5, c = -2 \)[/tex].
3. [tex]\( x^2 + 2x - 5 = 0 \)[/tex]
- Here, the coefficients are:
- [tex]\( a = 1, \)[/tex]
- [tex]\( b = 2, \)[/tex]
- [tex]\( c = -5. \)[/tex]
- These do not match with [tex]\( a = 4, b = 5, c = -2 \)[/tex].
4. [tex]\( x^2 - 5 = 0 \)[/tex]
- Here, the coefficients are:
- [tex]\( a = 1, \)[/tex]
- [tex]\( b = 0, \)[/tex]
- [tex]\( c = -5. \)[/tex]
- These do not match with [tex]\( a = 4, b = 5, c = -2 \)[/tex].
5. [tex]\( 4x^2 + 5x - 2 = 0 \)[/tex]
- Here, the coefficients are:
- [tex]\( a = 4, \)[/tex]
- [tex]\( b = 5, \)[/tex]
- [tex]\( c = -2. \)[/tex]
- These match perfectly with [tex]\( a = 4, b = 5, c = -2 \)[/tex].
Therefore, the correct labeling of [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex] to the standard form of the quadratic equation is given by:
[tex]\[ 4x^2 + 5x - 2 = 0. \][/tex]
So, the correct equation is:
- [tex]\( 4x^2 + 5x - 2 = 0 \)[/tex]
The label associated with this equation is number 5. Thus, the correct option is:
- Option 5
