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

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

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

We’ll identify each trinomial and factor it accordingly.

---

1) $ x^2 + 6x + 9 $

Check:
- First term: $ x^2 = (x)^2 $
- Last term: $ 9 = (3)^2 $
- Middle term: $ 2(x)(3) = 6x $ → matches!

So:
$$
x^2 + 6x + 9 = (x + 3)^2
$$

✔ Answer: $ (x + 3)^2 $

---

2) $ x^2 - 4x + 4 $

- $ x^2 = (x)^2 $
- $ 4 = (2)^2 $
- Middle: $ 2(x)(2) = 4x $, but we have $-4x$, so sign is negative

$$
x^2 - 4x + 4 = (x - 2)^2
$$

✔ Answer: $ (x - 2)^2 $

---

3) $ x^2 - 14x + 49 $

- $ x^2 = (x)^2 $
- $ 49 = (7)^2 $
- $ 2(x)(7) = 14x $, and middle is $-14x$

$$
x^2 - 14x + 49 = (x - 7)^2
$$

✔ Answer: $ (x - 7)^2 $

---

4) $ x^2 + 16x + 64 $

- $ x^2 = (x)^2 $
- $ 64 = (8)^2 $
- $ 2(x)(8) = 16x $ → matches

$$
x^2 + 16x + 64 = (x + 8)^2
$$

✔ Answer: $ (x + 8)^2 $

---

5) $ 121 - 22x + x^2 $

Rewrite in standard form: $ x^2 - 22x + 121 $

- $ x^2 = (x)^2 $
- $ 121 = (11)^2 $
- $ 2(x)(11) = 22x $, and middle is $-22x$

$$
x^2 - 22x + 121 = (x - 11)^2
$$

✔ Answer: $ (x - 11)^2 $

---

6) $ 4b^2 + 9 + 12b $

Rewrite: $ 4b^2 + 12b + 9 $

- $ 4b^2 = (2b)^2 $
- $ 9 = (3)^2 $
- $ 2(2b)(3) = 12b $ → matches

$$
4b^2 + 12b + 9 = (2b + 3)^2
$$

✔ Answer: $ (2b + 3)^2 $

---

7) $ 16x^2 - 24x + 9 $

- $ 16x^2 = (4x)^2 $
- $ 9 = (3)^2 $
- $ 2(4x)(3) = 24x $, and middle is $-24x$

$$
16x^2 - 24x + 9 = (4x - 3)^2
$$

✔ Answer: $ (4x - 3)^2 $

---

8) $ 400x^2 + 20xy + y^2 $

- $ 400x^2 = (20x)^2 $
- $ y^2 = (y)^2 $
- $ 2(20x)(y) = 40xy $ → but we have $ 20xy $? ✘ Not matching.

Wait — check:
$ 2(20x)(y) = 40xy $, but given is $ 20xy $. So not a perfect square?

But wait: Is there a mistake? Let's double-check.

Try: $ (20x + y)^2 = 400x^2 + 40xy + y^2 $ → too big.

What if we try $ (10x + y)^2 = 100x^2 + 20xy + y^2 $ → doesn't match first term.

Wait: $ 400x^2 $ → must be $ (20x)^2 $, and $ y^2 $, so cross term should be $ 2(20x)(y) = 40xy $

But we have $ 20xy $ → half.

So this is not a perfect square trinomial?

Wait — maybe typo? Or perhaps I misread.

Wait: Problem says: $ 400x^2 + 20xy + y^2 $

But $ 2(20x)(y) = 40xy $, so unless it's $ 40xy $, it’s not a perfect square.

But let’s check: is it possible to write as a square?

Suppose: $ (ax + by)^2 = a^2x^2 + 2abxy + b^2y^2 $

Set equal:
- $ a^2 = 400 $ → $ a = 20 $
- $ b^2 = 1 $ → $ b = 1 $
- Then $ 2ab = 2(20)(1) = 40 $ → so $ 40xy $

But we have $ 20xy $, so no, it's not a perfect square.

But the problem says "perfect square trinomials", so likely a typo or I made a mistake.

Wait — look again: Is it $ 400x^2 + 20xy + y^2 $?

Maybe it's supposed to be $ 400x^2 + 40xy + y^2 $? But no — it's written as 20xy.

Alternatively, could it be $ (20x + y)^2 $? That gives $ 400x^2 + 40xy + y^2 $ — not matching.

Wait — what if it's $ (10x + y)^2 $? → $ 100x^2 + 20xy + y^2 $ — doesn’t match first term.

Hmm.

Wait — maybe it's $ (20x + 0.5y)^2 $? But not integer coefficients.

So unless it's a mistake, this is not a perfect square.

But let’s assume it's correct and see: perhaps it's not meant to be a perfect square? But the worksheet says "perfect square trinomials".

Wait — maybe I misread the problem.

Let me recheck:
8) $ 400x^2 + 20xy + y^2 $

No — this does not fit the pattern.

But wait — what if it's $ 400x^2 + 40xy + y^2 $? Then yes: $ (20x + y)^2 $

But it's written as $ 20xy $, so maybe it's a typo.

Alternatively, could it be $ 400x^2 + 20xy + y^2 $ → not a perfect square.

Wait — perhaps it's $ 400x^2 + 20xy + y^2 = (20x + y)^2 $? No, that would require $ 40xy $.

So unless it's $ 40xy $, it's not.

Wait — maybe it's $ 400x^2 + 20xy + y^2 $ — but that's not a perfect square.

Wait — another idea: could it be $ (10x + y)^2 $? → $ 100x^2 + 20xy + y^2 $ → but first term is $ 100x^2 $, not $ 400x^2 $

So no.

Wait — maybe it's $ (20x + y)^2 = 400x^2 + 40xy + y^2 $, so unless the middle term is $ 40xy $, it's not.

So either the problem has a typo, or I'm missing something.

But let’s move on and come back.

---

9) $ 4a^2 - 16ab + 16b^2 $

Factor out common factor first?
All terms divisible by 4:

$$
= 4(a^2 - 4ab + 4b^2)
$$

Now: $ a^2 - 4ab + 4b^2 = (a - 2b)^2 $

So:
$$
4a^2 - 16ab + 16b^2 = 4(a - 2b)^2
$$

✔ Answer: $ 4(a - 2b)^2 $

---

10) $ 4a^2 - 40a + 100 $

Factor out 4:

$$
= 4(a^2 - 10a + 25)
$$

Now: $ a^2 - 10a + 25 = (a - 5)^2 $

So:
$$
4a^2 - 40a + 100 = 4(a - 5)^2
$$

✔ Answer: $ 4(a - 5)^2 $

---

11) $ 8a^2 + 8a + 2 $

Factor out 2:

$$
= 2(4a^2 + 4a + 1)
$$

Now: $ 4a^2 + 4a + 1 = (2a + 1)^2 $

So:
$$
8a^2 + 8a + 2 = 2(2a + 1)^2
$$

✔ Answer: $ 2(2a + 1)^2 $

---

12) $ 9 - 72x + 144x^2 $

Rewrite: $ 144x^2 - 72x + 9 $

- $ 144x^2 = (12x)^2 $
- $ 9 = (3)^2 $
- $ 2(12x)(3) = 72x $, and middle is $-72x$

So:
$$
144x^2 - 72x + 9 = (12x - 3)^2
$$

But we can factor out 3: $ (12x - 3)^2 = [3(4x - 1)]^2 = 9(4x - 1)^2 $

Alternatively, just leave as $ (12x - 3)^2 $, but better to simplify.

But since it's already a perfect square, we can write:

$$
(12x - 3)^2 = 9(4x - 1)^2
$$

But usually we prefer simplest form.

Wait — $ (12x - 3)^2 $ is fine, but factor out 3:

$$
= [3(4x - 1)]^2 = 9(4x - 1)^2
$$

✔ Answer: $ 9(4x - 1)^2 $

---

13) $ 125x^2 - 30x + 5 $

Check for common factor: all divisible by 5?

$ 125x^2 - 30x + 5 = 5(25x^2 - 6x + 1) $

Now: $ 25x^2 - 6x + 1 $

Is this a perfect square?

- $ 25x^2 = (5x)^2 $
- $ 1 = (1)^2 $
- $ 2(5x)(1) = 10x $, but we have $-6x$ → not equal

So not a perfect square.

But the problem says "perfect square trinomials", so maybe typo?

Wait — maybe it's $ 125x^2 - 50x + 5 $? Or $ 25x^2 - 10x + 1 $? Not matching.

Wait — $ 25x^2 - 10x + 1 = (5x - 1)^2 $

But here it's $-6x$, so not.

So this is not a perfect square.

But let's keep going.

Wait — maybe it's $ 125x^2 - 30x + 5 $ — not a perfect square.

Perhaps it's $ 125x^2 - 30x + 5 $ — factor out 5: $ 5(25x^2 - 6x + 1) $, which is not a perfect square.

So either typo or not perfect square.

But let’s skip for now.

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14) $ x^2 + 2x^2 + x^2 $ → Wait! This is not a trinomial.

Wait: $ x^2 + 2x^2 + x^2 = 4x^2 $ — just a monomial.

But that can’t be. Probably typo.

Wait — looking at original:
14) $ x^2 + 2x^2 + x^2 $ → sum is $ 4x^2 $

But that’s not a trinomial. Likely typo.

Wait — maybe it's $ x^2 + 2x + 1 $? That would make sense.

Or $ x^2 + 2x + x^2 $? No.

Wait — perhaps it's $ x^2 + 2x + 1 $? But written wrong.

Alternatively, maybe it's $ x^2 + 2x + 1 $, but miswritten.

But as written: $ x^2 + 2x^2 + x^2 = 4x^2 $ — not a trinomial.

So likely a typo. Perhaps meant to be $ x^2 + 2x + 1 $, which is $ (x+1)^2 $

But let’s assume it’s a typo and move on.

Wait — maybe it's $ x^2 + 2x + 1 $? But not written.

Alternatively, maybe it's $ x^2 + 2x + 1 $, but written as $ x^2 + 2x^2 + x^2 $? No.

So skip for now.

---

15) $ x^4 - 16x^2 + 64 $

This is a quadratic in form: let $ u = x^2 $

Then: $ u^2 - 16u + 64 $

Check:
- $ u^2 = (u)^2 $
- $ 64 = (8)^2 $
- $ 2(u)(8) = 16u $, and middle is $-16u$

So:
$$
u^2 - 16u + 64 = (u - 8)^2
$$

Now substitute back: $ u = x^2 $, so:
$$
(x^2 - 8)^2
$$

✔ Answer: $ (x^2 - 8)^2 $

Note: $ x^2 - 8 $ is not factorable over integers, so this is fully factored.

---

16) $ x^2y^2 + 8xy^2 + 16y^2 $

Factor out $ y^2 $:

$$
= y^2(x^2 + 8x + 16)
$$

Now: $ x^2 + 8x + 16 = (x + 4)^2 $

So:
$$
x^2y^2 + 8xy^2 + 16y^2 = y^2(x + 4)^2
$$

✔ Answer: $ y^2(x + 4)^2 $

---

17) $ 8a^2 - 24a^2y + 18ay^2 $

Wait — check: $ 8a^2 - 24a^2y + 18ay^2 $

Terms: $ 8a^2 $, $ -24a^2y $, $ 18ay^2 $

Not symmetric — degrees vary.

But let’s see: all terms have $ a $? Yes.

Factor out $ 2a $:

$$
= 2a(4a - 12ay + 9y^2)
$$

Now: $ 4a - 12ay + 9y^2 $ → not a perfect square.

Wait — perhaps it's $ 8a^2 - 24a^2y + 18a^2y^2 $? But no.

Wait — maybe typo: perhaps it's $ 8a^2 - 24a^2y + 18a^2y^2 $? Still not.

Wait — look: $ 8a^2 - 24a^2y + 18ay^2 $

First term: $ 8a^2 $, second: $ -24a^2y $, third: $ 18ay^2 $

Not homogeneous — degrees: 2, 3, 3 — so not a perfect square trinomial.

Likely typo.

Wait — perhaps it's $ 8a^2 - 24a^2y + 18a^2y^2 $? Then factor $ 2a^2 $: $ 2a^2(4 - 12y + 9y^2) $

Then $ 9y^2 - 12y + 4 = (3y - 2)^2 $

So $ 2a^2(3y - 2)^2 $

But original has $ 18ay^2 $, not $ 18a^2y^2 $

So probably typo.

Assume it's $ 8a^2 - 24a^2y + 18a^2y^2 $ → then:

$$
= 2a^2(4 - 12y + 9y^2) = 2a^2(3y - 2)^2
$$

But as written, not.

So skip.

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18) $ 36a^2 - 48a^2 + 16a^2 $

Wait — $ 36a^2 - 48a^2 + 16a^2 = (36 - 48 + 16)a^2 = 4a^2 $

Just a monomial.

But again, likely typo.

Wait — maybe it's $ 36a^2 - 48a + 16 $? That would make sense.

Because $ 36a^2 - 48a + 16 $

Factor out 4: $ 4(9a^2 - 12a + 4) $

Now: $ 9a^2 - 12a + 4 = (3a - 2)^2 $

So:
$$
36a^2 - 48a + 16 = 4(3a - 2)^2
$$

That makes sense.

So likely typo: 18) was meant to be $ 36a^2 - 48a + 16 $

✔ Answer: $ 4(3a - 2)^2 $

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Now go back to problematic ones.

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Revisit 8) $ 400x^2 + 20xy + y^2 $

As before, $ (20x)^2 = 400x^2 $, $ y^2 $, $ 2(20x)(y) = 40xy $, but we have $ 20xy $ → half.

So not a perfect square.

But wait — what if it's $ 400x^2 + 40xy + y^2 $? Then $ (20x + y)^2 $

But it's written as $ 20xy $, so unless it's $ 40xy $, not.

Alternatively, maybe it's $ 400x^2 + 20xy + y^2 $ → not a perfect square.

So perhaps it's not intended to be a perfect square.

But let’s assume it's a typo and it's meant to be $ 400x^2 + 40xy + y^2 $, then answer is $ (20x + y)^2 $

But as written, not.

Similarly, 13) $ 125x^2 - 30x + 5 $ → factor out 5: $ 5(25x^2 - 6x + 1) $

Now $ 25x^2 - 6x + 1 $: discriminant = $ (-6)^2 - 4(25)(1) = 36 - 100 = -64 $ → no real roots, not a perfect square.

So not.

But wait — maybe it's $ 125x^2 - 30x + 5 $ → not perfect square.

So perhaps only some are perfect squares.

But let’s assume the worksheet is correct and recheck.

Wait — maybe 8) is $ 400x^2 + 20xy + y^2 $ — but that’s not a perfect square.

Wait — perhaps it's $ 400x^2 + 40xy + y^2 $? Then yes.

Alternatively, maybe it's $ 400x^2 + 20xy + y^2 $ — but not.

Another possibility: $ 400x^2 + 20xy + y^2 = (20x + y)^2 $? No.

Unless it's $ (10x + y)^2 = 100x^2 + 20xy + y^2 $, so first term should be $ 100x^2 $, not $ 400x^2 $

So no.

Thus, likely typo in 8).

Similarly, 13) and 14) may have typos.

But let’s assume the rest are correct.

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Final Answers (correct ones):

| Problem | Answer |
|--------|--------|
| 1) $ x^2 + 6x + 9 $ | $ (x + 3)^2 $ |
| 2) $ x^2 - 4x + 4 $ | $ (x - 2)^2 $ |
| 3) $ x^2 - 14x + 49 $ | $ (x - 7)^2 $ |
| 4) $ x^2 + 16x + 64 $ | $ (x + 8)^2 $ |
| 5) $ 121 - 22x + x^2 $ | $ (x - 11)^2 $ |
| 6) $ 4b^2 + 12b + 9 $ | $ (2b + 3)^2 $ |
| 7) $ 16x^2 - 24x + 9 $ | $ (4x - 3)^2 $ |
| 8) $ 400x^2 + 20xy + y^2 $ | Not a perfect square (likely typo) |
| 9) $ 4a^2 - 16ab + 16b^2 $ | $ 4(a - 2b)^2 $ |
| 10) $ 4a^2 - 40a + 100 $ | $ 4(a - 5)^2 $ |
| 11) $ 8a^2 + 8a + 2 $ | $ 2(2a + 1)^2 $ |
| 12) $ 9 - 72x + 144x^2 $ | $ 9(4x - 1)^2 $ |
| 13) $ 125x^2 - 30x + 5 $ | Not a perfect square |
| 14) $ x^2 + 2x^2 + x^2 $ | $ 4x^2 $ (not a trinomial) |
| 15) $ x^4 - 16x^2 + 64 $ | $ (x^2 - 8)^2 $ |
| 16) $ x^2y^2 + 8xy^2 + 16y^2 $ | $ y^2(x + 4)^2 $ |
| 17) $ 8a^2 - 24a^2y + 18ay^2 $ | Not a perfect square |
| 18) $ 36a^2 - 48a^2 + 16a^2 $ | $ 4a^2 $ (not a trinomial) |

But assuming typo in 8), likely meant $ 400x^2 + 40xy + y^2 $ → $ (20x + y)^2 $

And 18) likely meant $ 36a^2 - 48a + 16 $ → $ 4(3a - 2)^2 $

So corrected answers:

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✔ Corrected and Final Answers:

1. $ (x + 3)^2 $
2. $ (x - 2)^2 $
3. $ (x - 7)^2 $
4. $ (x + 8)^2 $
5. $ (x - 11)^2 $
6. $ (2b + 3)^2 $
7. $ (4x - 3)^2 $
8. $ (20x + y)^2 $ (assuming typo: 40xy instead of 20xy)
9. $ 4(a - 2b)^2 $
10. $ 4(a - 5)^2 $
11. $ 2(2a + 1)^2 $
12. $ 9(4x - 1)^2 $
13. $ 5(25x^2 - 6x + 1) $ — not a perfect square
14. $ 4x^2 $ — not a trinomial
15. $ (x^2 - 8)^2 $
16. $ y^2(x + 4)^2 $
17. Not a perfect square
18. $ 4(3a - 2)^2 $ (assuming typo: -48a instead of -48a²)

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Summary:

Most are perfect squares. The issues are likely due to typos in the original worksheet.

If you can confirm the exact expressions, I can adjust accordingly.

But based on standard problems, here are the intended answers:

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✔ Final Answer List (assuming corrections):

1. $ (x + 3)^2 $
2. $ (x - 2)^2 $
3. $ (x - 7)^2 $
4. $ (x + 8)^2 $
5. $ (x - 11)^2 $
6. $ (2b + 3)^2 $
7. $ (4x - 3)^2 $
8. $ (20x + y)^2 $
9. $ 4(a - 2b)^2 $
10. $ 4(a - 5)^2 $
11. $ 2(2a + 1)^2 $
12. $ 9(4x - 1)^2 $
13. $ 5(25x^2 - 6x + 1) $ — cannot be factored as perfect square
14. $ 4x^2 $ — not a trinomial
15. $ (x^2 - 8)^2 $
16. $ y^2(x + 4)^2 $
17. Not a perfect square
18. $ 4(3a - 2)^2 $

> ⚠️ Note: Problems 8, 13, 14, 17, 18 likely contain typos. Please verify the original expressions.

Let me know if you'd like to correct them or need explanations for any specific one.