Perfect Square Trinomials.pdf - RHHS - Math - Free Printable

Let's solve each of these perfect square trinomial factoring problems step by step. A perfect square trinomial is a quadratic expression that can be written in the form:

$$
(a + b)^2 = a^2 + 2ab + b^2 \quad \text{or} \quad (a - b)^2 = a^2 - 2ab + b^2
$$

We’ll look for patterns where:
- The first and last terms are perfect squares.
- The middle term is twice the product of the square roots of the first and last terms.

---

1) $ 9n^2 - 180n + 900 $

- $ 9n^2 = (3n)^2 $
- $ 900 = 30^2 $
- Middle term: $ -180n = -2 \cdot (3n) \cdot 30 $

So:
$$
(3n - 30)^2
$$

✔ Answer: $ (3n - 30)^2 $

---

2) $ 64x^2 + 112x + 49 $

- $ 64x^2 = (8x)^2 $
- $ 49 = 7^2 $
- Middle term: $ 112x = 2 \cdot 8x \cdot 7 = 112x $

So:
$$
(8x + 7)^2
$$

✔ Answer: $ (8x + 7)^2 $

---

3) $ 512m^2 + 896m + 392 $

First, factor out GCF:
All coefficients divisible by 8?

- $ 512 ÷ 8 = 64 $
- $ 896 ÷ 8 = 112 $
- $ 392 ÷ 8 = 49 $

So:
$$
8(64m^2 + 112m + 49)
$$

Now check inside:
- $ 64m^2 = (8m)^2 $
- $ 49 = 7^2 $
- $ 112m = 2 \cdot 8m \cdot 7 $

So:
$$
8(8m + 7)^2
$$

✔ Answer: $ 8(8m + 7)^2 $

---

4) $ 4 + 12m + 9m^2 $

Rewrite in standard form: $ 9m^2 + 12m + 4 $

- $ 9m^2 = (3m)^2 $
- $ 4 = 2^2 $
- $ 12m = 2 \cdot 3m \cdot 2 $

So:
$$
(3m + 2)^2
$$

✔ Answer: $ (3m + 2)^2 $

---

5) $ 81a^2 - 18a + 1 $

- $ 81a^2 = (9a)^2 $
- $ 1 = 1^2 $
- $ -18a = -2 \cdot 9a \cdot 1 $

So:
$$
(9a - 1)^2
$$

✔ Answer: $ (9a - 1)^2 $

---

6) $ 810 + 180a + 10a^2 $

Rewrite: $ 10a^2 + 180a + 810 $

Factor out GCF: all divisible by 10?

- $ 10a^2 $
- $ 180a $
- $ 810 $

Yes: $ 10(a^2 + 18a + 81) $

Now: $ a^2 + 18a + 81 $

- $ a^2 = (a)^2 $
- $ 81 = 9^2 $
- $ 18a = 2 \cdot a \cdot 9 $

So:
$$
10(a + 9)^2
$$

✔ Answer: $ 10(a + 9)^2 $

---

7) $ 392 + 896x + 512x^2 $

Rewrite: $ 512x^2 + 896x + 392 $

Factor out GCF: all divisible by 8?

- $ 512 ÷ 8 = 64 $
- $ 896 ÷ 8 = 112 $
- $ 392 ÷ 8 = 49 $

So:
$$
8(64x^2 + 112x + 49)
$$

Now: $ 64x^2 = (8x)^2 $, $ 49 = 7^2 $, $ 112x = 2 \cdot 8x \cdot 7 $

So:
$$
8(8x + 7)^2
$$

✔ Answer: $ 8(8x + 7)^2 $

---

8) $ x^2 - 16x + 64 $

- $ x^2 = (x)^2 $
- $ 64 = 8^2 $
- $ -16x = -2 \cdot x \cdot 8 $

So:
$$
(x - 8)^2
$$

✔ Answer: $ (x - 8)^2 $

---

9) $ 10 - 140n + 490n^2 $

Rewrite: $ 490n^2 - 140n + 10 $

Factor out GCF: 10

- $ 490n^2 ÷ 10 = 49n^2 $
- $ -140n ÷ 10 = -14n $
- $ 10 ÷ 10 = 1 $

So:
$$
10(49n^2 - 14n + 1)
$$

Now: $ 49n^2 = (7n)^2 $, $ 1 = 1^2 $, $ -14n = -2 \cdot 7n \cdot 1 $

So:
$$
10(7n - 1)^2
$$

✔ Answer: $ 10(7n - 1)^2 $

---

10) $ 64 - 48n + 9n^2 $

Rewrite: $ 9n^2 - 48n + 64 $

- $ 9n^2 = (3n)^2 $
- $ 64 = 8^2 $
- $ -48n = -2 \cdot 3n \cdot 8 $

So:
$$
(3n - 8)^2
$$

✔ Answer: $ (3n - 8)^2 $

---

11) $ 49y^2 - 70y + 25 $

- $ 49y^2 = (7y)^2 $
- $ 25 = 5^2 $
- $ -70y = -2 \cdot 7y \cdot 5 $

So:
$$
(7y - 5)^2
$$

✔ Answer: $ (7y - 5)^2 $

---

12) $ 64n^2 - 32n + 4 $

Factor out GCF: 4

- $ 64n^2 ÷ 4 = 16n^2 $
- $ -32n ÷ 4 = -8n $
- $ 4 ÷ 4 = 1 $

So:
$$
4(16n^2 - 8n + 1)
$$

Now: $ 16n^2 = (4n)^2 $, $ 1 = 1^2 $, $ -8n = -2 \cdot 4n \cdot 1 $

So:
$$
4(4n - 1)^2
$$

✔ Answer: $ 4(4n - 1)^2 $

---

13) $ 16x^2 + 40x + 25 $

- $ 16x^2 = (4x)^2 $
- $ 25 = 5^2 $
- $ 40x = 2 \cdot 4x \cdot 5 $

So:
$$
(4x + 5)^2
$$

✔ Answer: $ (4x + 5)^2 $

---

14) $ 1 + 2x + x^2 $

Rewrite: $ x^2 + 2x + 1 $

- $ x^2 = (x)^2 $
- $ 1 = 1^2 $
- $ 2x = 2 \cdot x \cdot 1 $

So:
$$
(x + 1)^2
$$

✔ Answer: $ (x + 1)^2 $

---

15) $ 25m^2 + 90m + 81 $

- $ 25m^2 = (5m)^2 $
- $ 81 = 9^2 $
- $ 90m = 2 \cdot 5m \cdot 9 $

So:
$$
(5m + 9)^2
$$

✔ Answer: $ (5m + 9)^2 $

---

16) $ 96v^2 + 48v + 6 $

Factor out GCF: 6

- $ 96v^2 ÷ 6 = 16v^2 $
- $ 48v ÷ 6 = 8v $
- $ 6 ÷ 6 = 1 $

So:
$$
6(16v^2 + 8v + 1)
$$

Now: $ 16v^2 = (4v)^2 $, $ 1 = 1^2 $, $ 8v = 2 \cdot 4v \cdot 1 $

So:
$$
6(4v + 1)^2
$$

✔ Answer: $ 6(4v + 1)^2 $

---

17) $ 250 - 900w + 810w^2 $

Rewrite: $ 810w^2 - 900w + 250 $

Factor out GCF: 10

- $ 810w^2 ÷ 10 = 81w^2 $
- $ -900w ÷ 10 = -90w $
- $ 250 ÷ 10 = 25 $

So:
$$
10(81w^2 - 90w + 25)
$$

Now: $ 81w^2 = (9w)^2 $, $ 25 = 5^2 $, $ -90w = -2 \cdot 9w \cdot 5 $

So:
$$
10(9w - 5)^2
$$

✔ Answer: $ 10(9w - 5)^2 $

---

18) $ 640v^2 + 1120v + 490 $

Factor out GCF: 10

- $ 640v^2 ÷ 10 = 64v^2 $
- $ 1120v ÷ 10 = 112v $
- $ 490 ÷ 10 = 49 $

So:
$$
10(64v^2 + 112v + 49)
$$

Now: $ 64v^2 = (8v)^2 $, $ 49 = 7^2 $, $ 112v = 2 \cdot 8v \cdot 7 $

So:
$$
10(8v + 7)^2
$$

✔ Answer: $ 10(8v + 7)^2 $

---

19) $ 288k^2 + 672k + 392 $

Factor out GCF: let’s see — all even? Yes.

Try dividing by 8:

- $ 288 ÷ 8 = 36 $
- $ 672 ÷ 8 = 84 $
- $ 392 ÷ 8 = 49 $

So:
$$
8(36k^2 + 84k + 49)
$$

Now: $ 36k^2 = (6k)^2 $, $ 49 = 7^2 $, $ 84k = 2 \cdot 6k \cdot 7 $

So:
$$
8(6k + 7)^2
$$

✔ Answer: $ 8(6k + 7)^2 $

---

20) $ 1 - 12x + 36x^2 $

Rewrite: $ 36x^2 - 12x + 1 $

- $ 36x^2 = (6x)^2 $
- $ 1 = 1^2 $
- $ -12x = -2 \cdot 6x \cdot 1 $

So:
$$
(6x - 1)^2
$$

✔ Answer: $ (6x - 1)^2 $

---

21) $ 162x^2 - 36x + 2 $

Factor out GCF: 2

- $ 162x^2 ÷ 2 = 81x^2 $
- $ -36x ÷ 2 = -18x $
- $ 2 ÷ 2 = 1 $

So:
$$
2(81x^2 - 18x + 1)
$$

Now: $ 81x^2 = (9x)^2 $, $ 1 = 1^2 $, $ -18x = -2 \cdot 9x \cdot 1 $

So:
$$
2(9x - 1)^2
$$

✔ Answer: $ 2(9x - 1)^2 $

---

22) $ 36n^2 - 60n + 25 $

- $ 36n^2 = (6n)^2 $
- $ 25 = 5^2 $
- $ -60n = -2 \cdot 6n \cdot 5 $

So:
$$
(6n - 5)^2
$$

✔ Answer: $ (6n - 5)^2 $

---

23) $ 9n^2 - 48n + 64 $

- $ 9n^2 = (3n)^2 $
- $ 64 = 8^2 $
- $ -48n = -2 \cdot 3n \cdot 8 $

So:
$$
(3n - 8)^2
$$

✔ Answer: $ (3n - 8)^2 $

---

24) $ 162x^2 - 180x + 50 $

Factor out GCF: 2

- $ 162x^2 ÷ 2 = 81x^2 $
- $ -180x ÷ 2 = -90x $
- $ 50 ÷ 2 = 25 $

So:
$$
2(81x^2 - 90x + 25)
$$

Now: $ 81x^2 = (9x)^2 $, $ 25 = 5^2 $, $ -90x = -2 \cdot 9x \cdot 5 $

So:
$$
2(9x - 5)^2
$$

✔ Answer: $ 2(9x - 5)^2 $

---

25) $ 25x^2 + 70x + 49 $

- $ 25x^2 = (5x)^2 $
- $ 49 = 7^2 $
- $ 70x = 2 \cdot 5x \cdot 7 $

So:
$$
(5x + 7)^2
$$

✔ Answer: $ (5x + 7)^2 $

---

26) $ 128r^2 + 160r + 50 $

Factor out GCF: 2

- $ 128r^2 ÷ 2 = 64r^2 $
- $ 160r ÷ 2 = 80r $
- $ 50 ÷ 2 = 25 $

So:
$$
2(64r^2 + 80r + 25)
$$

Now: $ 64r^2 = (8r)^2 $, $ 25 = 5^2 $, $ 80r = 2 \cdot 8r \cdot 5 $

So:
$$
2(8r + 5)^2
$$

✔ Answer: $ 2(8r + 5)^2 $

---

27) $ 9 + 18p + 9p^2 $

Rewrite: $ 9p^2 + 18p + 9 $

Factor out GCF: 9

- $ 9p^2 ÷ 9 = p^2 $
- $ 18p ÷ 9 = 2p $
- $ 9 ÷ 9 = 1 $

So:
$$
9(p^2 + 2p + 1)
$$

Now: $ p^2 + 2p + 1 = (p + 1)^2 $

So:
$$
9(p + 1)^2
$$

✔ Answer: $ 9(p + 1)^2 $

---

28) $ 4a^2 + 4a + 1 $

- $ 4a^2 = (2a)^2 $
- $ 1 = 1^2 $
- $ 4a = 2 \cdot 2a \cdot 1 $

So:
$$
(2a + 1)^2
$$

✔ Answer: $ (2a + 1)^2 $

---

29) $ 324m^2 - 720m + 400 $

- $ 324m^2 = (18m)^2 $
- $ 400 = 20^2 $
- $ -720m = -2 \cdot 18m \cdot 20 $? Let's check:

$ 2 \cdot 18m \cdot 20 = 720m $ → yes

So:
$$
(18m - 20)^2
$$

But we can factor out common factor from binomial?

Check if both terms divisible by 2: $ 18m - 20 = 2(9m - 10) $

So:
$$
(18m - 20)^2 = [2(9m - 10)]^2 = 4(9m - 10)^2
$$

But unless asked to simplify further, $ (18m - 20)^2 $ is acceptable.

But better to write as:
$$
(18m - 20)^2 = 4(9m - 10)^2
$$

Wait: actually, it's already a perfect square. But we can factor out GCF from original?

Original: $ 324m^2 - 720m + 400 $

GCF? All divisible by 4?

- $ 324 ÷ 4 = 81 $
- $ 720 ÷ 4 = 180 $
- $ 400 ÷ 4 = 100 $

So:
$$
4(81m^2 - 180m + 100)
$$

Now: $ 81m^2 = (9m)^2 $, $ 100 = 10^2 $, $ -180m = -2 \cdot 9m \cdot 10 $

So:
$$
4(9m - 10)^2
$$

✔ Answer: $ 4(9m - 10)^2 $

---

30) $ 175 + 490n + 343n^2 $

Rewrite: $ 343n^2 + 490n + 175 $

Factor out GCF: 7?

- $ 343 ÷ 7 = 49 $
- $ 490 ÷ 7 = 70 $
- $ 175 ÷ 7 = 25 $

So:
$$
7(49n^2 + 70n + 25)
$$

Now: $ 49n^2 = (7n)^2 $, $ 25 = 5^2 $, $ 70n = 2 \cdot 7n \cdot 5 $

So:
$$
7(7n + 5)^2
$$

✔ Answer: $ 7(7n + 5)^2 $

---

## ✔ Final Answers:

| Problem | Factored Form |
|--------|----------------|
| 1) | $ (3n - 30)^2 $ |
| 2) | $ (8x + 7)^2 $ |
| 3) | $ 8(8m + 7)^2 $ |
| 4) | $ (3m + 2)^2 $ |
| 5) | $ (9a - 1)^2 $ |
| 6) | $ 10(a + 9)^2 $ |
| 7) | $ 8(8x + 7)^2 $ |
| 8) | $ (x - 8)^2 $ |
| 9) | $ 10(7n - 1)^2 $ |
| 10) | $ (3n - 8)^2 $ |
| 11) | $ (7y - 5)^2 $ |
| 12) | $ 4(4n - 1)^2 $ |
| 13) | $ (4x + 5)^2 $ |
| 14) | $ (x + 1)^2 $ |
| 15) | $ (5m + 9)^2 $ |
| 16) | $ 6(4v + 1)^2 $ |
| 17) | $ 10(9w - 5)^2 $ |
| 18) | $ 10(8v + 7)^2 $ |
| 19) | $ 8(6k + 7)^2 $ |
| 20) | $ (6x - 1)^2 $ |
| 21) | $ 2(9x - 1)^2 $ |
| 22) | $ (6n - 5)^2 $ |
| 23) | $ (3n - 8)^2 $ |
| 24) | $ 2(9x - 5)^2 $ |
| 25) | $ (5x + 7)^2 $ |
| 26) | $ 2(8r + 5)^2 $ |
| 27) | $ 9(p + 1)^2 $ |
| 28) | $ (2a + 1)^2 $ |
| 29) | $ 4(9m - 10)^2 $ |
| 30) | $ 7(7n + 5)^2 $ |

Let me know if you'd like this formatted as a downloadable PDF or need explanations for any specific one!