To determine the correct values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] for the given quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] with roots [tex]\( r_1 \)[/tex] and [tex]\( r_2 \)[/tex] such that [tex]\( r_1 + r_2 = -2.5 \)[/tex] and [tex]\( r_1r_2 = 1.5 \)[/tex], follow these steps:
1. Identify the relationships:
- If [tex]\( r_1 \)[/tex] and [tex]\( r_2 \)[/tex] are the roots of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], then by Vieta's formulas:
- The sum of the roots [tex]\( r_1 + r_2 \)[/tex] is given by [tex]\( -\frac{b}{a} \)[/tex].
- The product of the roots [tex]\( r_1r_2 \)[/tex] is given by [tex]\( \frac{c}{a} \)[/tex].
2. Given conditions:
- [tex]\( r_1 + r_2 = -2.5 \)[/tex]
- [tex]\( r_1r_2 = 1.5 \)[/tex]
3. Relate these conditions to [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex]:
- Therefore, [tex]\( -\frac{b}{a} = -2.5 \)[/tex]
- This simplifies to [tex]\( \frac{b}{a} = 2.5 \)[/tex], or [tex]\( b = 2.5a \)[/tex].
- And [tex]\( \frac{c}{a} = 1.5 \)[/tex]
- This simplifies to [tex]\( c = 1.5a \)[/tex].
4. Check each option:
Option A: [tex]\( a = 1 \)[/tex], [tex]\( b = 6 \)[/tex], [tex]\( c = 5 \)[/tex]
- Here, [tex]\(\frac{b}{a} = \frac{6}{1} = 6 \neq 2.5 \)[/tex]
- So, this option is incorrect.
Option B: [tex]\( a = 1 \)[/tex], [tex]\( b = 5 \)[/tex], [tex]\( c = 6 \)[/tex]
- Here, [tex]\(\frac{b}{a} = \frac{5}{1} = 5 \neq 2.5 \)[/tex]
- So, this option is incorrect.
Option C: [tex]\( a = 2 \)[/tex], [tex]\( b = 3 \)[/tex], [tex]\( c = 5 \)[/tex]
- Here, [tex]\(\frac{b}{a} = \frac{3}{2} = 1.5 \neq 2.5 \)[/tex]
- So, this option is incorrect.
Option D: [tex]\( a = 2 \)[/tex], [tex]\( b = 5 \)[/tex], [tex]\( c = 3 \)[/tex]
- Here, [tex]\(\frac{b}{a} = \frac{5}{2} = 2.5 \)[/tex]
- And [tex]\(\frac{c}{a} = \frac{3}{2} = 1.5 \)[/tex]
These values for [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] satisfy both conditions:
- [tex]\( r_1 + r_2 = -2.5 \)[/tex]
- [tex]\( r_1r_2 = 1.5 \)[/tex]
Therefore, the correct values for [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are [tex]\( \boxed{2, 5, 3} \)[/tex].
