Adding and Subtracting and Simplifying Quadratic Expressions with ... - Free Printable

Problem Overview:

The task involves simplifying quadratic expressions by adding and subtracting terms. Each expression contains variables raised to the power of 2 (quadratic terms), linear terms, and constant terms. The goal is to combine like terms to simplify each expression.

Solution Explanation:

To solve these problems, follow these steps:
1. Distribute coefficients: If there are coefficients outside parentheses, distribute them to each term inside the parentheses.
2. Combine like terms: Group and add/subtract terms with the same variable and power.
3. Simplify: Write the final expression in standard form, arranging terms from highest to lowest degree.

Let's go through each problem step by step.

---

Problem 1:

\[
-8(5t^2 + 2t + 2) + 9(-6t^2 + 2t - 8)
\]

#### Step 1: Distribute the coefficients
\[
-8(5t^2 + 2t + 2) = -40t^2 - 16t - 16
\]
\[
9(-6t^2 + 2t - 8) = -54t^2 + 18t - 72
\]

#### Step 2: Combine like terms
\[
(-40t^2 - 54t^2) + (-16t + 18t) + (-16 - 72)
\]
\[
-94t^2 + 2t - 88
\]

#### Final Answer:
\[
\boxed{-94t^2 + 2t - 88}
\]

---

Problem 2:

\[
(7m^2 + 9m - 8) - 5(-7m^2 - 4m - 1)
\]

#### Step 1: Distribute the coefficients
\[
7m^2 + 9m - 8
\]
\[
-5(-7m^2 - 4m - 1) = 35m^2 + 20m + 5
\]

#### Step 2: Combine like terms
\[
(7m^2 + 35m^2) + (9m + 20m) + (-8 + 5)
\]
\[
42m^2 + 29m - 3
\]

#### Final Answer:
\[
\boxed{42m^2 + 29m - 3}
\]

---

Problem 3:

\[
2(-3w^2 - 7w + 2) + 9(4w^2 + 2w + 7)
\]

#### Step 1: Distribute the coefficients
\[
2(-3w^2 - 7w + 2) = -6w^2 - 14w + 4
\]
\[
9(4w^2 + 2w + 7) = 36w^2 + 18w + 63
\]

#### Step 2: Combine like terms
\[
(-6w^2 + 36w^2) + (-14w + 18w) + (4 + 63)
\]
\[
30w^2 + 4w + 67
\]

#### Final Answer:
\[
\boxed{30w^2 + 4w + 67}
\]

---

Problem 4:

\[
2(-w^2 + w - 4) + 9(9w^2 - w + 9)
\]

#### Step 1: Distribute the coefficients
\[
2(-w^2 + w - 4) = -2w^2 + 2w - 8
\]
\[
9(9w^2 - w + 9) = 81w^2 - 9w + 81
\]

#### Step 2: Combine like terms
\[
(-2w^2 + 81w^2) + (2w - 9w) + (-8 + 81)
\]
\[
79w^2 - 7w + 73
\]

#### Final Answer:
\[
\boxed{79w^2 - 7w + 73}
\]

---

Problem 5:

\[
(2v^2 + 5v + 8) + (-9v^2 + 3v - 6)
\]

#### Step 1: Combine like terms
\[
(2v^2 - 9v^2) + (5v + 3v) + (8 - 6)
\]
\[
-7v^2 + 8v + 2
\]

#### Final Answer:
\[
\boxed{-7v^2 + 8v + 2}
\]

---

Problem 6:

\[
(-7r^2 - 5r + 8) + 5(8r^2 - 4r + 1)
\]

#### Step 1: Distribute the coefficients
\[
-7r^2 - 5r + 8
\]
\[
5(8r^2 - 4r + 1) = 40r^2 - 20r + 5
\]

#### Step 2: Combine like terms
\[
(-7r^2 + 40r^2) + (-5r - 20r) + (8 + 5)
\]
\[
33r^2 - 25r + 13
\]

#### Final Answer:
\[
\boxed{33r^2 - 25r + 13}
\]

---

Problem 7:

\[
(-7b^2 - 3b - 9) - 7(8b^2 + 9b + 6)
\]

#### Step 1: Distribute the coefficients
\[
-7b^2 - 3b - 9
\]
\[
-7(8b^2 + 9b + 6) = -56b^2 - 63b - 42
\]

#### Step 2: Combine like terms
\[
(-7b^2 - 56b^2) + (-3b - 63b) + (-9 - 42)
\]
\[
-63b^2 - 66b - 51
\]

#### Final Answer:
\[
\boxed{-63b^2 - 66b - 51}
\]

---

Problem 8:

\[
(-4h^2 + 5h - 8) - (-h^2 - 3h + 2)
\]

#### Step 1: Distribute the negative sign
\[
-4h^2 + 5h - 8 + h^2 + 3h - 2
\]

#### Step 2: Combine like terms
\[
(-4h^2 + h^2) + (5h + 3h) + (-8 - 2)
\]
\[
-3h^2 + 8h - 10
\]

#### Final Answer:
\[
\boxed{-3h^2 + 8h - 10}
\]

---

Problem 9:

\[
(9b^2 + 4b - 2) - 4(9b^2 + 2)
\]

#### Step 1: Distribute the coefficients
\[
9b^2 + 4b - 2
\]
\[
-4(9b^2 + 2) = -36b^2 - 8
\]

#### Step 2: Combine like terms
\[
(9b^2 - 36b^2) + (4b) + (-2 - 8)
\]
\[
-27b^2 + 4b - 10
\]

#### Final Answer:
\[
\boxed{-27b^2 + 4b - 10}
\]

---

Problem 10:

\[
(-9z^2 + 8z - 5) - 7(8z^2 + 9z + 4)
\]

#### Step 1: Distribute the coefficients
\[
-9z^2 + 8z - 5
\]
\[
-7(8z^2 + 9z + 4) = -56z^2 - 63z - 28
\]

#### Step 2: Combine like terms
\[
(-9z^2 - 56z^2) + (8z - 63z) + (-5 - 28)
\]
\[
-65z^2 - 55z - 33
\]

#### Final Answer:
\[
\boxed{-65z^2 - 55z - 33}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ -94t^2 + 2t - 88 \\
2. & \ 42m^2 + 29m - 3 \\
3. & \ 30w^2 + 4w + 67 \\
4. & \ 79w^2 - 7w + 73 \\
5. & \ -7v^2 + 8v + 2 \\
6. & \ 33r^2 - 25r + 13 \\
7. & \ -63b^2 - 66b - 51 \\
8. & \ -3h^2 + 8h - 10 \\
9. & \ -27b^2 + 4b - 10 \\
10. & \ -65z^2 - 55z - 33
\end{aligned}
}
\]