Honors Algebra 2 worksheet with practice problems on expressions, formulas, and real-world applications.

Problem: Solve the given algebraic expressions and word problems.

#### Expressions and Formulas

1. Find the value of each expression.

- 1. \( 3d - 7 = -11 \)
\[
3d - 7 = -11
\]
Add 7 to both sides:
\[
3d = -4
\]
Divide by 3:
\[
d = -\frac{4}{3}
\]

- 2. \( 12 - 4^2 \)
\[
12 - 4^2 = 12 - 16 = -4
\]

- 3. \( 1 + 2 - 3(4) + 2 \)
\[
1 + 2 - 3(4) + 2 = 1 + 2 - 12 + 2 = 5 - 12 = -7
\]

- 4. \( 12 - [20 - 2(6)^2 + 3 \times 2^2] \)
Simplify inside the brackets:
\[
2(6)^2 = 2 \times 36 = 72, \quad 3 \times 2^2 = 3 \times 4 = 12
\]
\[
20 - 72 + 12 = -40
\]
Now subtract:
\[
12 - (-40) = 12 + 40 = 52
\]

- 5. \( 20 + (8 - 3) + 5^2(3) \)
Simplify inside the parentheses and exponents:
\[
8 - 3 = 5, \quad 5^2 = 25, \quad 25 \times 3 = 75
\]
\[
20 + 5 + 75 = 100
\]

- 6. \( (-2)^3 - (3)(8) + (5)(19) \)
Simplify each term:
\[
(-2)^3 = -8, \quad (3)(8) = 24, \quad (5)(19) = 95
\]
\[
-8 - 24 + 95 = -32 + 95 = 63
\]

- 7. \( 17 - [-83 - (34 - 17 - 11)] \)
Simplify inside the innermost parentheses:
\[
34 - 17 - 11 = 34 - 28 = 6
\]
Now simplify further:
\[
-83 - 6 = -89
\]
Finally:
\[
17 - (-89) = 17 + 89 = 106
\]

- 8. \( [4(5 - 3) + 24 - 8] + 16 \)
Simplify inside the brackets:
\[
5 - 3 = 2, \quad 4(2) = 8, \quad 8 + 24 - 8 = 24
\]
Now add:
\[
24 + 16 = 40
\]

- 9. \( \frac{1}{4}(6 - 4^2) \)
Simplify inside the parentheses:
\[
4^2 = 16, \quad 6 - 16 = -10
\]
Now multiply:
\[
\frac{1}{4}(-10) = -\frac{10}{4} = -\frac{5}{2}
\]

- 10. \( \frac{1}{4}[-3 + 6(-3)] \)
Simplify inside the brackets:
\[
6(-3) = -18, \quad -3 + (-18) = -21
\]
Now multiply:
\[
\frac{1}{4}(-21) = -\frac{21}{4}
\]

- 11. \( \frac{-8(3) + 37}{6} \)
Simplify the numerator:
\[
-8(3) = -24, \quad -24 + 37 = 13
\]
Now divide:
\[
\frac{13}{6}
\]

- 12. \( \frac{-6^2 - 9}{-1^3 + 6(-9)} \)
Simplify the numerator and denominator:
\[
-6^2 = -36, \quad -1^3 = -1, \quad 6(-9) = -54
\]
\[
\text{Numerator: } -36 - 9 = -45
\]
\[
\text{Denominator: } -1 - 54 = -55
\]
Now divide:
\[
\frac{-45}{-55} = \frac{45}{55} = \frac{9}{11}
\]

#### Evaluate each expression if \( u = \frac{2}{3} \), \( b = -8 \), \( d = -2 \), \( e = 3 \), and \( v = \frac{1}{3} \).

13. \( c^2b - d \)
\[
c^2b - d = (3)^2(-8) - (-2) = 9(-8) + 2 = -72 + 2 = -70
\]

14. \( e + d^6 \)
\[
e + d^6 = 3 + (-2)^6 = 3 + 64 = 67
\]

15. \( \frac{ab}{c} + a^2 \)
\[
\frac{ab}{c} + a^2 = \frac{\left(\frac{2}{3}\right)(-8)}{3} + \left(\frac{2}{3}\right)^2 = \frac{-16/3}{3} + \frac{4}{9} = -\frac{16}{9} + \frac{4}{9} = -\frac{12}{9} = -\frac{4}{3}
\]

16. \( \frac{d^2 - e}{ac} \)
\[
\frac{d^2 - e}{ac} = \frac{(-2)^2 - 3}{\left(\frac{2}{3}\right)(3)} = \frac{4 - 3}{2} = \frac{1}{2}
\]

17. \( (b - dv)c^2 \)
\[
(b - dv)c^2 = (-8 - (-2)\left(\frac{1}{3}\right))(3)^2 = \left(-8 + \frac{2}{3}\right)(9) = \left(-\frac{24}{3} + \frac{2}{3}\right)(9) = \left(-\frac{22}{3}\right)(9) = -66
\]

18. \( av^2 - b^2v \)
\[
av^2 - b^2v = \left(\frac{2}{3}\right)\left(\frac{1}{3}\right)^2 - (-8)^2\left(\frac{1}{3}\right) = \left(\frac{2}{3}\right)\left(\frac{1}{9}\right) - 64\left(\frac{1}{3}\right) = \frac{2}{27} - \frac{64}{3} = \frac{2}{27} - \frac{576}{27} = -\frac{574}{27}
\]

19. \( -(dta) + (v - d|2|) \)
\[
-(dta) + (v - d|2|) = -((-2)\left(\frac{2}{3}\right)a) + \left(\frac{1}{3} - (-2)(2)\right) = \frac{4a}{3} + \left(\frac{1}{3} + 4\right) = \frac{4a}{3} + \frac{13}{3}
\]

20. \( \frac{a^3}{e^2} - \frac{e}{a} \)
\[
\frac{a^3}{e^2} - \frac{e}{a} = \frac{\left(\frac{2}{3}\right)^3}{3^2} - \frac{3}{\frac{2}{3}} = \frac{\frac{8}{27}}{9} - \frac{9}{2} = \frac{8}{243} - \frac{9}{2} = \frac{8}{243} - \frac{2187}{243} = -\frac{2179}{243}
\]

21. \( |9b - c| \)
\[
|9b - c| = |9(-8) - 3| = |-72 - 3| = |-75| = 75
\]

22. \( |3ab^3 - (d^3 - v)| \)
\[
|3ab^3 - (d^3 - v)| = \left|3\left(\frac{2}{3}\right)(-8)^3 - \left((-2)^3 - \frac{1}{3}\right)\right| = \left|2(-512) - \left(-8 - \frac{1}{3}\right)\right| = \left|-1024 - \left(-\frac{24}{3} - \frac{1}{3}\right)\right| = \left|-1024 + \frac{25}{3}\right| = \left|-\frac{3072}{3} + \frac{25}{3}\right| = \left|-\frac{3047}{3}\right| = \frac{3047}{3}
\]

#### Word Problems

23. Temperature Conversion
The formula is \( F = \frac{9}{5}C + 32 \). For \( C = -40 \):
\[
F = \frac{9}{5}(-40) + 32 = -72 + 32 = -40
\]

24. Physics: Height of an Object
The formula is \( h = 120t - 0.1t^2 \). For \( t = 6 \):
\[
h = 120(6) - 0.1(6)^2 = 720 - 0.1(36) = 720 - 3.6 = 716.4
\]

25. Agriculture: Predicted Profit
The formula is \( P = 26x - 0.01x^2 - 240 \). For \( x = 50 \):
\[
P = 26(50) - 0.01(50)^2 - 240 = 1300 - 0.01(2500) - 240 = 1300 - 25 - 240 = 1035
\]

Final Answers:

\[
\boxed{
\begin{aligned}
&1. d = -\frac{4}{3}, \quad 2. -4, \quad 3. -7, \quad 4. 52, \quad 5. 100, \quad 6. 63, \quad 7. 106, \quad 8. 40, \quad 9. -\frac{5}{2}, \quad 10. -\frac{21}{4}, \\
&11. \frac{13}{6}, \quad 12. \frac{9}{11}, \quad 13. -70, \quad 14. 67, \quad 15. -\frac{4}{3}, \quad 16. \frac{1}{2}, \quad 17. -66, \quad 18. -\frac{574}{27}, \\
&19. \frac{4a}{3} + \frac{13}{3}, \quad 20. -\frac{2179}{243}, \quad 21. 75, \quad 22. \frac{3047}{3}, \quad 23. -40, \quad 24. 716.4, \quad 25. 1035.
\end{aligned}
}
\]