Factoring Perfect Square Trinomials Worksheet - Fill and Sign ... - Free Printable

Problem Description:

The task is to factor each of the given quadratic expressions completely. The expressions are listed in two columns, and we need to solve them step by step.

---

Solution:

#### Column 1:

1. Expression: \( u^2 + 6u + 9 \)
- This is a perfect square trinomial.
- Recall the formula for a perfect square: \( (a + b)^2 = a^2 + 2ab + b^2 \).
- Here, \( a = u \) and \( b = 3 \), so:
\[
u^2 + 6u + 9 = (u + 3)^2
\]
- Answer: \( (u + 3)^2 \)

2. Expression: \( u^2 - 4u + 4 \)
- This is also a perfect square trinomial.
- Using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[
u^2 - 4u + 4 = (u - 2)^2
\]
- Answer: \( (u - 2)^2 \)

3. Expression: \( u^2 - 14u + 49 \)
- Again, this is a perfect square trinomial.
- Using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[
u^2 - 14u + 49 = (u - 7)^2
\]
- Answer: \( (u - 7)^2 \)

4. Expression: \( u^2 + 18u + 81 \)
- This is a perfect square trinomial.
- Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[
u^2 + 18u + 81 = (u + 9)^2
\]
- Answer: \( (u + 9)^2 \)

5. Expression: \( 121 - 22u + u^2 \)
- Rearrange the terms to match the standard form:
\[
u^2 - 22u + 121
\]
- This is a perfect square trinomial.
- Using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[
u^2 - 22u + 121 = (u - 11)^2
\]
- Answer: \( (u - 11)^2 \)

6. Expression: \( 9u^2 + 6uv + v^2 \)
- This is a perfect square trinomial involving two variables.
- Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[
9u^2 + 6uv + v^2 = (3u + v)^2
\]
- Answer: \( (3u + v)^2 \)

7. Expression: \( 100x^2 - 20xy + y^2 \)
- This is a perfect square trinomial involving two variables.
- Using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[
100x^2 - 20xy + y^2 = (10x - y)^2
\]
- Answer: \( (10x - y)^2 \)

8. Expression: \( 49x^2 - 28xy + 4y^2 \)
- This is a perfect square trinomial involving two variables.
- Using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[
49x^2 - 28xy + 4y^2 = (7x - 2y)^2
\]
- Answer: \( (7x - 2y)^2 \)

9. Expression: \( 4u^2 - 9uv + 9v^2 \)
- This expression does not factor into a perfect square trinomial.
- However, it can be factored using the method of grouping or recognizing it as a sum/difference of squares. In this case, it does not simplify further into a perfect square.
- Answer: \( 4u^2 - 9uv + 9v^2 \) (no further simplification)

---

#### Column 2:

10. Expression: \( 8u^2 + 8u + 2 \)
- Factor out the greatest common factor (GCF):
\[
8u^2 + 8u + 2 = 2(4u^2 + 4u + 1)
\]
- The expression inside the parentheses is a perfect square trinomial:
\[
4u^2 + 4u + 1 = (2u + 1)^2
\]
- Therefore:
\[
8u^2 + 8u + 2 = 2(2u + 1)^2
\]
- Answer: \( 2(2u + 1)^2 \)

11. Expression: \( 3u^2 + 18u + 27 \)
- Factor out the GCF:
\[
3u^2 + 18u + 27 = 3(u^2 + 6u + 9)
\]
- The expression inside the parentheses is a perfect square trinomial:
\[
u^2 + 6u + 9 = (u + 3)^2
\]
- Therefore:
\[
3u^2 + 18u + 27 = 3(u + 3)^2
\]
- Answer: \( 3(u + 3)^2 \)

12. Expression: \( 9 - 72u + 144u^2 \)
- Rearrange the terms:
\[
144u^2 - 72u + 9
\]
- Factor out the GCF:
\[
144u^2 - 72u + 9 = 9(16u^2 - 8u + 1)
\]
- The expression inside the parentheses is a perfect square trinomial:
\[
16u^2 - 8u + 1 = (4u - 1)^2
\]
- Therefore:
\[
9 - 72u + 144u^2 = 9(4u - 1)^2
\]
- Answer: \( 9(4u - 1)^2 \)

13. Expression: \( 125u^2 - 30u + 5 \)
- Factor out the GCF:
\[
125u^2 - 30u + 5 = 5(25u^2 - 6u + 1)
\]
- The expression inside the parentheses is a perfect square trinomial:
\[
25u^2 - 6u + 1 = (5u - 1)^2
\]
- Therefore:
\[
125u^2 - 30u + 5 = 5(5u - 1)^2
\]
- Answer: \( 5(5u - 1)^2 \)

14. Expression: \( u^2 - 2u^3 + u^4 \)
- Rearrange the terms:
\[
u^4 - 2u^3 + u^2
\]
- Factor out the GCF:
\[
u^4 - 2u^3 + u^2 = u^2(u^2 - 2u + 1)
\]
- The expression inside the parentheses is a perfect square trinomial:
\[
u^2 - 2u + 1 = (u - 1)^2
\]
- Therefore:
\[
u^4 - 2u^3 + u^2 = u^2(u - 1)^2
\]
- Answer: \( u^2(u - 1)^2 \)

15. Expression: \( u^5 - 16u^3 + 64u \)
- Factor out the GCF:
\[
u^5 - 16u^3 + 64u = u(u^4 - 16u^2 + 64)
\]
- The expression inside the parentheses is a perfect square trinomial:
\[
u^4 - 16u^2 + 64 = (u^2 - 8)^2
\]
- Therefore:
\[
u^5 - 16u^3 + 64u = u(u^2 - 8)^2
\]
- Answer: \( u(u^2 - 8)^2 \)

16. Expression: \( x^2y^2 - 6xy^3 + 9y^4 \)
- Factor out the GCF:
\[
x^2y^2 - 6xy^3 + 9y^4 = y^2(x^2 - 6xy + 9y^2)
\]
- The expression inside the parentheses is a perfect square trinomial:
\[
x^2 - 6xy + 9y^2 = (x - 3y)^2
\]
- Therefore:
\[
x^2y^2 - 6xy^3 + 9y^4 = y^2(x - 3y)^2
\]
- Answer: \( y^2(x - 3y)^2 \)

17. Expression: \( 8x^2 - 24xy^2 + 18y^4 \)
- Factor out the GCF:
\[
8x^2 - 24xy^2 + 18y^4 = 2(4x^2 - 12xy^2 + 9y^4)
\]
- The expression inside the parentheses is a perfect square trinomial:
\[
4x^2 - 12xy^2 + 9y^4 = (2x - 3y^2)^2
\]
- Therefore:
\[
8x^2 - 24xy^2 + 18y^4 = 2(2x - 3y^2)^2
\]
- Answer: \( 2(2x - 3y^2)^2 \)

18. Expression: \( 36u^2 - 48uv + 16v^2 \)
- Factor out the GCF:
\[
36u^2 - 48uv + 16v^2 = 4(9u^2 - 12uv + 4v^2)
\]
- The expression inside the parentheses is a perfect square trinomial:
\[
9u^2 - 12uv + 4v^2 = (3u - 2v)^2
\]
- Therefore:
\[
36u^2 - 48uv + 16v^2 = 4(3u - 2v)^2
\]
- Answer: \( 4(3u - 2v)^2 \)

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & (u + 3)^2 \\
2. & (u - 2)^2 \\
3. & (u - 7)^2 \\
4. & (u + 9)^2 \\
5. & (u - 11)^2 \\
6. & (3u + v)^2 \\
7. & (10x - y)^2 \\
8. & (7x - 2y)^2 \\
9. & 4u^2 - 9uv + 9v^2 \\
10. & 2(2u + 1)^2 \\
11. & 3(u + 3)^2 \\
12. & 9(4u - 1)^2 \\
13. & 5(5u - 1)^2 \\
14. & u^2(u - 1)^2 \\
15. & u(u^2 - 8)^2 \\
16. & y^2(x - 3y)^2 \\
17. & 2(2x - 3y^2)^2 \\
18. & 4(3u - 2v)^2 \\
\end{array}
}
\]