To determine which choices are solutions to the given equation
[tex]\[
x^2 - 3x = -\frac{5}{4}
\][/tex]
we can check each of the possible values for [tex]\(x\)[/tex] and see if they satisfy the equation.
1. Checking [tex]\( x = 2 \)[/tex]:
[tex]\[
x^2 - 3x = 2^2 - 3 \cdot 2 = 4 - 6 = -2
\][/tex]
Since [tex]\(-2 \neq -\frac{5}{4}\)[/tex], [tex]\( x = 2 \)[/tex] is not a solution.
2. Checking [tex]\( x = 2.5 \)[/tex]:
[tex]\[
x^2 - 3x = 2.5^2 - 3 \cdot 2.5 = 6.25 - 7.5 = -1.25
\][/tex]
Since [tex]\(-1.25 = -\frac{5}{4}\)[/tex], [tex]\( x = 2.5 \)[/tex] is a solution.
3. Checking [tex]\( x = 0.5 \)[/tex]:
[tex]\[
x^2 - 3x = 0.5^2 - 3 \cdot 0.5 = 0.25 - 1.5 = -1.25
\][/tex]
Since [tex]\(-1.25 = -\frac{5}{4}\)[/tex], [tex]\( x = 0.5 \)[/tex] is a solution.
4. Checking [tex]\( x = -1 \)[/tex]:
[tex]\[
x^2 - 3x = (-1)^2 - 3 \cdot (-1) = 1 + 3 = 4
\][/tex]
Since [tex]\( 4 \neq -\frac{5}{4}\)[/tex], [tex]\( x = -1 \)[/tex] is not a solution.
Therefore, the values that satisfy the equation [tex]\( x^2 - 3x = -\frac{5}{4} \)[/tex] are:
- [tex]\( x = 2.5 \)[/tex]
- [tex]\( x = 0.5 \)[/tex]
Thus, the correct choices are [tex]\( \boxed{B \text{ and } C} \)[/tex].
