This Assignment 1.6 worksheet provides 24 practice problems for students to solve various algebraic equations using factoring methods.

To solve the given equations by factoring, we will go through each equation step by step. Here is the detailed solution for each problem:

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Equation 1: \( x^2 + 4x = 0 \)

1. Factor out the greatest common factor (GCF), which is \( x \):
\[
x(x + 4) = 0
\]
2. Use the zero-product property:
\[
x = 0 \quad \text{or} \quad x + 4 = 0
\]
3. Solve for \( x \):
\[
x = 0 \quad \text{or} \quad x = -4
\]

Solution: \( x = 0 \) or \( x = -4 \)

---

Equation 2: \( x^2 + 8x + 15 = 0 \)

1. Factor the quadratic expression:
\[
x^2 + 8x + 15 = (x + 3)(x + 5)
\]
2. Set each factor equal to zero:
\[
x + 3 = 0 \quad \text{or} \quad x + 5 = 0
\]
3. Solve for \( x \):
\[
x = -3 \quad \text{or} \quad x = -5
\]

Solution: \( x = -3 \) or \( x = -5 \)

---

Equation 3: \( x^2 - 9x + 18 = 0 \)

1. Factor the quadratic expression:
\[
x^2 - 9x + 18 = (x - 3)(x - 6)
\]
2. Set each factor equal to zero:
\[
x - 3 = 0 \quad \text{or} \quad x - 6 = 0
\]
3. Solve for \( x \):
\[
x = 3 \quad \text{or} \quad x = 6
\]

Solution: \( x = 3 \) or \( x = 6 \)

---

Equation 4: \( x^2 - 16 = 0 \)

1. Recognize that this is a difference of squares:
\[
x^2 - 16 = (x - 4)(x + 4)
\]
2. Set each factor equal to zero:
\[
x - 4 = 0 \quad \text{or} \quad x + 4 = 0
\]
3. Solve for \( x \):
\[
x = 4 \quad \text{or} \quad x = -4
\]

Solution: \( x = 4 \) or \( x = -4 \)

---

Equation 5: \( x^2 - 6x = 0 \)

1. Factor out the GCF, which is \( x \):
\[
x(x - 6) = 0
\]
2. Use the zero-product property:
\[
x = 0 \quad \text{or} \quad x - 6 = 0
\]
3. Solve for \( x \):
\[
x = 0 \quad \text{or} \quad x = 6
\]

Solution: \( x = 0 \) or \( x = 6 \)

---

Equation 6: \( x^2 + 7x + 10 = 0 \)

1. Factor the quadratic expression:
\[
x^2 + 7x + 10 = (x + 2)(x + 5)
\]
2. Set each factor equal to zero:
\[
x + 2 = 0 \quad \text{or} \quad x + 5 = 0
\]
3. Solve for \( x \):
\[
x = -2 \quad \text{or} \quad x = -5
\]

Solution: \( x = -2 \) or \( x = -5 \)

---

Equation 7: \( x^2 - 11x + 28 = 0 \)

1. Factor the quadratic expression:
\[
x^2 - 11x + 28 = (x - 4)(x - 7)
\]
2. Set each factor equal to zero:
\[
x - 4 = 0 \quad \text{or} \quad x - 7 = 0
\]
3. Solve for \( x \):
\[
x = 4 \quad \text{or} \quad x = 7
\]

Solution: \( x = 4 \) or \( x = 7 \)

---

Equation 8: \( x^2 - 25 = 0 \)

1. Recognize that this is a difference of squares:
\[
x^2 - 25 = (x - 5)(x + 5)
\]
2. Set each factor equal to zero:
\[
x - 5 = 0 \quad \text{or} \quad x + 5 = 0
\]
3. Solve for \( x \):
\[
x = 5 \quad \text{or} \quad x = -5
\]

Solution: \( x = 5 \) or \( x = -5 \)

---

Equation 9: \( x^2 - 8x = 0 \)

1. Factor out the GCF, which is \( x \):
\[
x(x - 8) = 0
\]
2. Use the zero-product property:
\[
x = 0 \quad \text{or} \quad x - 8 = 0
\]
3. Solve for \( x \):
\[
x = 0 \quad \text{or} \quad x = 8
\]

Solution: \( x = 0 \) or \( x = 8 \)

---

Equation 10: \( x^2 + 9x + 18 = 0 \)

1. Factor the quadratic expression:
\[
x^2 + 9x + 18 = (x + 3)(x + 6)
\]
2. Set each factor equal to zero:
\[
x + 3 = 0 \quad \text{or} \quad x + 6 = 0
\]
3. Solve for \( x \):
\[
x = -3 \quad \text{or} \quad x = -6
\]

Solution: \( x = -3 \) or \( x = -6 \)

---

Equation 11: \( x^2 - 13x + 42 = 0 \)

1. Factor the quadratic expression:
\[
x^2 - 13x + 42 = (x - 6)(x - 7)
\]
2. Set each factor equal to zero:
\[
x - 6 = 0 \quad \text{or} \quad x - 7 = 0
\]
3. Solve for \( x \):
\[
x = 6 \quad \text{or} \quad x = 7
\]

Solution: \( x = 6 \) or \( x = 7 \)

---

Equation 12: \( x^2 - 36 = 0 \)

1. Recognize that this is a difference of squares:
\[
x^2 - 36 = (x - 6)(x + 6)
\]
2. Set each factor equal to zero:
\[
x - 6 = 0 \quad \text{or} \quad x + 6 = 0
\]
3. Solve for \( x \):
\[
x = 6 \quad \text{or} \quad x = -6
\]

Solution: \( x = 6 \) or \( x = -6 \)

---

Equation 13: \( x^2 - 10x = 0 \)

1. Factor out the GCF, which is \( x \):
\[
x(x - 10) = 0
\]
2. Use the zero-product property:
\[
x = 0 \quad \text{or} \quad x - 10 = 0
\]
3. Solve for \( x \):
\[
x = 0 \quad \text{or} \quad x = 10
\]

Solution: \( x = 0 \) or \( x = 10 \)

---

Equation 14: \( x^2 + 11x + 28 = 0 \)

1. Factor the quadratic expression:
\[
x^2 + 11x + 28 = (x + 4)(x + 7)
\]
2. Set each factor equal to zero:
\[
x + 4 = 0 \quad \text{or} \quad x + 7 = 0
\]
3. Solve for \( x \):
\[
x = -4 \quad \text{or} \quad x = -7
\]

Solution: \( x = -4 \) or \( x = -7 \)

---

Equation 15: \( x^2 - 15x + 56 = 0 \)

1. Factor the quadratic expression:
\[
x^2 - 15x + 56 = (x - 7)(x - 8)
\]
2. Set each factor equal to zero:
\[
x - 7 = 0 \quad \text{or} \quad x - 8 = 0
\]
3. Solve for \( x \):
\[
x = 7 \quad \text{or} \quad x = 8
\]

Solution: \( x = 7 \) or \( x = 8 \)

---

Equation 16: \( x^2 - 49 = 0 \)

1. Recognize that this is a difference of squares:
\[
x^2 - 49 = (x - 7)(x + 7)
\]
2. Set each factor equal to zero:
\[
x - 7 = 0 \quad \text{or} \quad x + 7 = 0
\]
3. Solve for \( x \):
\[
x = 7 \quad \text{or} \quad x = -7
\]

Solution: \( x = 7 \) or \( x = -7 \)

---

Equation 17: \( x^2 - 12x = 0 \)

1. Factor out the GCF, which is \( x \):
\[
x(x - 12) = 0
\]
2. Use the zero-product property:
\[
x = 0 \quad \text{or} \quad x - 12 = 0
\]
3. Solve for \( x \):
\[
x = 0 \quad \text{or} \quad x = 12
\]

Solution: \( x = 0 \) or \( x = 12 \)

---

Equation 18: \( x^2 + 12x + 35 = 0 \)

1. Factor the quadratic expression:
\[
x^2 + 12x + 35 = (x + 5)(x + 7)
\]
2. Set each factor equal to zero:
\[
x + 5 = 0 \quad \text{or} \quad x + 7 = 0
\]
3. Solve for \( x \):
\[
x = -5 \quad \text{or} \quad x = -7
\]

Solution: \( x = -5 \) or \( x = -7 \)

---

Equation 19: \( x^2 - 17x + 72 = 0 \)

1. Factor the quadratic expression:
\[
x^2 - 17x + 72 = (x - 8)(x - 9)
\]
2. Set each factor equal to zero:
\[
x - 8 = 0 \quad \text{or} \quad x - 9 = 0
\]
3. Solve for \( x \):
\[
x = 8 \quad \text{or} \quad x = 9
\]

Solution: \( x = 8 \) or \( x = 9 \)

---

Equation 20: \( x^2 - 64 = 0 \)

1. Recognize that this is a difference of squares:
\[
x^2 - 64 = (x - 8)(x + 8)
\]
2. Set each factor equal to zero:
\[
x - 8 = 0 \quad \text{or} \quad x + 8 = 0
\]
3. Solve for \( x \):
\[
x = 8 \quad \text{or} \quad x = -8
\]

Solution: \( x = 8 \) or \( x = -8 \)

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Final Answer:

\[
\boxed{
\begin{aligned}
&1. \ x = 0 \ \text{or} \ x = -4 \\
&2. \ x = -3 \ \text{or} \ x = -5 \\
&3. \ x = 3 \ \text{or} \ x = 6 \\
&4. \ x = 4 \ \text{or} \ x = -4 \\
&5. \ x = 0 \ \text{or} \ x = 6 \\
&6. \ x = -2 \ \text{or} \ x = -5 \\
&7. \ x = 4 \ \text{or} \ x = 7 \\
&8. \ x = 5 \ \text{or} \ x = -5 \\
&9. \ x = 0 \ \text{or} \ x = 8 \\
&10. \ x = -3 \ \text{or} \ x = -6 \\
&11. \ x = 6 \ \text{or} \ x = 7 \\
&12. \ x = 6 \ \text{or} \ x = -6 \\
&13. \ x = 0 \ \text{or} \ x = 10 \\
&14. \ x = -4 \ \text{or} \ x = -7 \\
&15. \ x = 7 \ \text{or} \ x = 8 \\
&16. \ x = 7 \ \text{or} \ x = -7 \\
&17. \ x = 0 \ \text{or} \ x = 12 \\
&18. \ x = -5 \ \text{or} \ x = -7 \\
&19. \ x = 8 \ \text{or} \ x = 9 \\
&20. \ x = 8 \ \text{or} \ x = -8 \\
\end{aligned}
}
\]