Practice worksheet for solving quadratic equations by factoring, illustrated with a hatching chick.

Problem: Solve the quadratic equations by factoring.

The task involves solving 10 quadratic equations using the method of factoring. Let's solve each equation step by step.

---

#### 1. \( m^2 - 65m = 0 \)

- Step 1: Factor out the greatest common factor (GCF), which is \( m \):
\[
m(m - 65) = 0
\]

- Step 2: Use the zero-product property:
\[
m = 0 \quad \text{or} \quad m - 65 = 0
\]
Solving for \( m \):
\[
m = 0 \quad \text{or} \quad m = 65
\]

- Solution:
\[
\boxed{m = 0, 65}
\]

---

#### 2. \( x^2 + 13x + 27 = 0 \)

- Step 1: Look for two numbers that multiply to \( 27 \) and add to \( 13 \). These numbers are \( 9 \) and \( 4 \).

- Step 2: Rewrite the quadratic expression using these numbers:
\[
x^2 + 13x + 27 = (x + 9)(x + 3)
\]

- Step 3: Set each factor equal to zero:
\[
(x + 9)(x + 3) = 0
\]
\[
x + 9 = 0 \quad \text{or} \quad x + 3 = 0
\]
Solving for \( x \):
\[
x = -9 \quad \text{or} \quad x = -3
\]

- Solution:
\[
\boxed{x = -9, -3}
\]

---

#### 3. \( k^2 - 12k = 0 \)

- Step 1: Factor out the GCF, which is \( k \):
\[
k(k - 12) = 0
\]

- Step 2: Use the zero-product property:
\[
k = 0 \quad \text{or} \quad k - 12 = 0
\]
Solving for \( k \):
\[
k = 0 \quad \text{or} \quad k = 12
\]

- Solution:
\[
\boxed{k = 0, 12}
\]

---

#### 4. \( b^2 + 12b + 35 = 0 \)

- Step 1: Look for two numbers that multiply to \( 35 \) and add to \( 12 \). These numbers are \( 7 \) and \( 5 \).

- Step 2: Rewrite the quadratic expression using these numbers:
\[
b^2 + 12b + 35 = (b + 7)(b + 5)
\]

- Step 3: Set each factor equal to zero:
\[
(b + 7)(b + 5) = 0
\]
\[
b + 7 = 0 \quad \text{or} \quad b + 5 = 0
\]
Solving for \( b \):
\[
b = -7 \quad \text{or} \quad b = -5
\]

- Solution:
\[
\boxed{b = -7, -5}
\]

---

#### 5. \( x^2 - 19x = 0 \)

- Step 1: Factor out the GCF, which is \( x \):
\[
x(x - 19) = 0
\]

- Step 2: Use the zero-product property:
\[
x = 0 \quad \text{or} \quad x - 19 = 0
\]
Solving for \( x \):
\[
x = 0 \quad \text{or} \quad x = 19
\]

- Solution:
\[
\boxed{x = 0, 19}
\]

---

#### 6. \( x^2 + 3x - 4x = 0 \)

- Step 1: Simplify the equation:
\[
x^2 - x = 0
\]

- Step 2: Factor out the GCF, which is \( x \):
\[
x(x - 1) = 0
\]

- Step 3: Use the zero-product property:
\[
x = 0 \quad \text{or} \quad x - 1 = 0
\]
Solving for \( x \):
\[
x = 0 \quad \text{or} \quad x = 1
\]

- Solution:
\[
\boxed{x = 0, 1}
\]

---

#### 7. \( x^2 + 9x = 0 \)

- Step 1: Factor out the GCF, which is \( x \):
\[
x(x + 9) = 0
\]

- Step 2: Use the zero-product property:
\[
x = 0 \quad \text{or} \quad x + 9 = 0
\]
Solving for \( x \):
\[
x = 0 \quad \text{or} \quad x = -9
\]

- Solution:
\[
\boxed{x = 0, -9}
\]

---

#### 8. \( x^2 - 10x^2 + 9 = 0 \)

- Step 1: Simplify the equation:
\[
-9x^2 + 9 = 0
\]

- Step 2: Factor out the GCF, which is \( -9 \):
\[
-9(x^2 - 1) = 0
\]

- Step 3: Recognize that \( x^2 - 1 \) is a difference of squares:
\[
x^2 - 1 = (x - 1)(x + 1)
\]
So the equation becomes:
\[
-9(x - 1)(x + 1) = 0
\]

- Step 4: Set each factor equal to zero:
\[
x - 1 = 0 \quad \text{or} \quad x + 1 = 0
\]
Solving for \( x \):
\[
x = 1 \quad \text{or} \quad x = -1
\]

- Solution:
\[
\boxed{x = 1, -1}
\]

---

#### 9. \( n^2 + 21n = 0 \)

- Step 1: Factor out the GCF, which is \( n \):
\[
n(n + 21) = 0
\]

- Step 2: Use the zero-product property:
\[
n = 0 \quad \text{or} \quad n + 21 = 0
\]
Solving for \( n \):
\[
n = 0 \quad \text{or} \quad n = -21
\]

- Solution:
\[
\boxed{n = 0, -21}
\]

---

#### 10. \( y^2 - 10y + 21 = 0 \)

- Step 1: Look for two numbers that multiply to \( 21 \) and add to \( -10 \). These numbers are \( -7 \) and \( -3 \).

- Step 2: Rewrite the quadratic expression using these numbers:
\[
y^2 - 10y + 21 = (y - 7)(y - 3)
\]

- Step 3: Set each factor equal to zero:
\[
(y - 7)(y - 3) = 0
\]
\[
y - 7 = 0 \quad \text{or} \quad y - 3 = 0
\]
Solving for \( y \):
\[
y = 7 \quad \text{or} \quad y = 3
\]

- Solution:
\[
\boxed{y = 7, 3}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \quad m = 0, 65 \\
2. & \quad x = -9, -3 \\
3. & \quad k = 0, 12 \\
4. & \quad b = -7, -5 \\
5. & \quad x = 0, 19 \\
6. & \quad x = 0, 1 \\
7. & \quad x = 0, -9 \\
8. & \quad x = 1, -1 \\
9. & \quad n = 0, -21 \\
10. & \quad y = 7, 3 \\
\end{aligned}
}
\]