Given a quadratic equation [tex]\( a x^2 + b x + c \)[/tex], we are asked to determine the relationship between the coefficient [tex]\( b \)[/tex] and the constant term [tex]\( c \)[/tex]. Let's go through each of the conditions provided and identify which one matches the given relationships:
1. Condition 1: [tex]\( c \)[/tex] is 15 times that of [tex]\( b \)[/tex].
- If [tex]\( b = 2 \)[/tex]:
- [tex]\( c = 15 \times b \)[/tex]
- [tex]\( c = 15 \times 2 = 30 \)[/tex]
- Thus, in this case, [tex]\( c = 30 \)[/tex] when [tex]\( b = 2 \)[/tex].
2. Condition 2: [tex]\( c \)[/tex] is double [tex]\( b \)[/tex].
- If [tex]\( b = 2 \)[/tex]:
- [tex]\( c = 2 \times b \)[/tex]
- [tex]\( c = 2 \times 2 = 4 \)[/tex]
- Thus, in this case, [tex]\( c = 4 \)[/tex] when [tex]\( b = 2 \)[/tex].
3. Condition 3: [tex]\( c \)[/tex] is the square of half of [tex]\( b \)[/tex].
- If [tex]\( b = 2 \)[/tex]:
- [tex]\( c = \left( \frac{b}{2} \right)^2 \)[/tex]
- [tex]\( c = \left( \frac{2}{2} \right)^2 = 1 \)[/tex]
- Thus, in this case, [tex]\( c = 1 \)[/tex] when [tex]\( b = 2 \)[/tex].
4. Condition 4: [tex]\( c \)[/tex] is the square of [tex]\( b \)[/tex].
- If [tex]\( b = 2 \)[/tex]:
- [tex]\( c = b^2 \)[/tex]
- [tex]\( c = 2^2 = 4 \)[/tex]
- Thus, in this case, [tex]\( c = 4 \)[/tex] when [tex]\( b = 2 \)[/tex].
Now, comparing the results:
- For Condition 1: When [tex]\( b = 2 \)[/tex], [tex]\( c = 30 \)[/tex].
- For Condition 2: When [tex]\( b = 2 \)[/tex], [tex]\( c = 4 \)[/tex].
- For Condition 3: When [tex]\( b = 2 \)[/tex], [tex]\( c = 1 \)[/tex].
- For Condition 4: When [tex]\( b = 2 \)[/tex], [tex]\( c = 4 \)[/tex].
Given the results, we see:
- [tex]\( c = 30 \)[/tex]
- [tex]\( c = 4 \)[/tex]
- [tex]\( c = 1.0 \)[/tex]
- [tex]\( c = 4 \)[/tex]
The conditions that correspond correctly to our determined values are the second and fourth options:
- [tex]\( c \)[/tex] is double [tex]\( b \)[/tex].
- [tex]\( c \)[/tex] is the square of [tex]\( b \)[/tex].
Therefore, the correct relationships are:
1) [tex]\( c \)[/tex] is double [tex]\( b \)[/tex].
2) [tex]\( c \)[/tex] is the square of [tex]\( b \)[/tex].
