Let's solve it step by step:
### Part (i)
We need to evaluate the expression [tex]\(-\frac{2}{3} x^2 + 5x - 2\)[/tex] for [tex]\( x = -3 \)[/tex].
1. Substitute [tex]\( x = -3 \)[/tex] into the expression:
[tex]\[
-\frac{2}{3}(-3)^2 + 5(-3) - 2
\][/tex]
2. Calculate [tex]\((-3)^2\)[/tex]:
[tex]\[
(-3)^2 = 9
\][/tex]
3. Substitute [tex]\( 9 \)[/tex] back into the expression:
[tex]\[
-\frac{2}{3} \cdot 9 + 5(-3) - 2
\][/tex]
4. Calculate [tex]\(-\frac{2}{3} \cdot 9\)[/tex]:
[tex]\[
-\frac{2}{3} \cdot 9 = -6
\][/tex]
5. Calculate [tex]\( 5(-3) \)[/tex]:
[tex]\[
5 \cdot -3 = -15
\][/tex]
6. Substitute these values back into the expression:
[tex]\[
-6 - 15 - 2
\][/tex]
7. Simplify the expression:
[tex]\[
-6 - 15 = -21
\][/tex]
[tex]\[
-21 - 2 = -23
\][/tex]
So, the value for part (i) is:
[tex]\[
-23.0
\][/tex]
### Part (ii)
We need to evaluate the expression [tex]\(\frac{3a + 2b - 5}{a + b}\)[/tex] for [tex]\(a = 2\)[/tex] and [tex]\(b = -1\)[/tex].
1. Substitute [tex]\( a = 2 \)[/tex] and [tex]\( b = -1 \)[/tex] into the expression:
[tex]\[
\frac{3 \cdot 2 + 2 \cdot (-1) - 5}{2 + (-1)}
\][/tex]
2. Calculate [tex]\( 3 \cdot 2 \)[/tex]:
[tex]\[
3 \cdot 2 = 6
\][/tex]
3. Calculate [tex]\( 2 \cdot (-1) \)[/tex]:
[tex]\[
2 \cdot (-1) = -2
\][/tex]
4. Substitute these values back into the expression:
[tex]\[
\frac{6 - 2 - 5}{2 - 1}
\][/tex]
5. Simplify the numerator:
[tex]\[
6 - 2 = 4
\][/tex]
[tex]\[
4 - 5 = -1
\][/tex]
6. Simplify the denominator:
[tex]\[
2 - 1 = 1
\][/tex]
7. Divide the numerator by the denominator:
[tex]\[
\frac{-1}{1} = -1
\][/tex]
So, the value for part (ii) is:
[tex]\[
-1.0
\][/tex]
Therefore, the final answers are:
- (i) [tex]\(-23.0\)[/tex]
- (ii) [tex]\(-1.0\)[/tex]
