Factoring Trinomial Squares with Leading Coefficient of 1 worksheet - Free Printable

Since I can't view or access images directly, I’ll help you solve the entire set of problems listed in your text — factoring trinomial squares with a leading coefficient of 1. These are quadratic expressions in the form $ x^2 + bx + c $, and we factor them into two binomials: $ (x + m)(x + n) $, where $ m \cdot n = c $ and $ m + n = b $.

Let’s go through each problem step by step.

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1) $ x^2 + 7x + 6 $

We need two numbers that multiply to 6 and add to 7:
- $ 6 \times 1 = 6 $, $ 6 + 1 = 7 $ → yes!
$$
(x + 6)(x + 1)
$$

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2) $ x^2 - 7x + 6 $

Numbers that multiply to 6, add to -7:
- $ -6 \times -1 = 6 $, $ -6 + (-1) = -7 $
$$
(x - 6)(x - 1)
$$

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3) $ p^2 - 4p - 45 $

Multiply to -45, add to -4:
Try: $ -9 \times 5 = -45 $, $ -9 + 5 = -4 $
$$
(p - 9)(p + 5)
$$

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4) $ b^2 - 8b + 12 $

Multiply to 12, add to -8:
- $ -6 \times -2 = 12 $, $ -6 + (-2) = -8 $
$$
(b - 6)(b - 2)
$$

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5) $ x^2 - 7x - 8 $

Multiply to -8, add to -7:
- $ -8 \times 1 = -8 $, $ -8 + 1 = -7 $
$$
(x - 8)(x + 1)
$$

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6) $ k^2 + 5k - 5 $

Try factors of $-5$: $ 5 \times -1 = -5 $, $ 5 + (-1) = 4 $ → no
$ -5 \times 1 = -5 $, $ -5 + 1 = -4 $ → no
No integers work → Prime (not factorable over integers)

> ✘ Cannot be factored using integers.

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7) $ a^2 + 14a + 48 $

Multiply to 48, add to 14:
- $ 6 \times 8 = 48 $, $ 6 + 8 = 14 $
$$
(a + 6)(a + 8)
$$

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8) $ n^2 - 15n + 50 $

Multiply to 50, add to -15:
- $ -10 \times -5 = 50 $, $ -10 + (-5) = -15 $
$$
(n - 10)(n - 5)
$$

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9) $ p^2 + 4p + 4 $

Perfect square trinomial:
- $ (p + 2)^2 $
$$
(p + 2)^2
$$

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10) $ x^2 - 6x - 27 $

Multiply to -27, add to -6:
- $ -9 \times 3 = -27 $, $ -9 + 3 = -6 $
$$
(x - 9)(x + 3)
$$

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11) $ r^2 - r + 4 $

Multiply to 4, add to -1:
- $ 1 \times 4 = 4 $, $ 1 + 4 = 5 $
- $ -1 \times -4 = 4 $, $ -1 + (-4) = -5 $
- No pair adds to -1 → Prime

> ✘ Not factorable over integers.

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12) $ x^2 + x - 72 $

Multiply to -72, add to 1:
- $ 9 \times -8 = -72 $, $ 9 + (-8) = 1 $
$$
(x + 9)(x - 8)
$$

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13) $ x^2 - 8x - 9 $

Multiply to -9, add to -8:
- $ -9 \times 1 = -9 $, $ -9 + 1 = -8 $
$$
(x - 9)(x + 1)
$$

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14) $ p^2 + 12p + 36 $

Perfect square:
- $ (p + 6)^2 $
$$
(p + 6)^2
$$

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15) $ n^2 - 7n - 18 $

Multiply to -18, add to -7:
- $ -9 \times 2 = -18 $, $ -9 + 2 = -7 $
$$
(n - 9)(n + 2)
$$

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16) $ m^2 + 11m + 28 $

Multiply to 28, add to 11:
- $ 7 \times 4 = 28 $, $ 7 + 4 = 11 $
$$
(m + 7)(m + 4)
$$

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17) $ m^2 + 12m + 27 $

Multiply to 27, add to 12:
- $ 9 \times 3 = 27 $, $ 9 + 3 = 12 $
$$
(m + 9)(m + 3)
$$

---

18) $ n^2 - 8n - 72 $

Multiply to -72, add to -8:
- $ -12 \times 6 = -72 $, $ -12 + 6 = -6 $
- $ -18 \times 4 = -72 $, $ -18 + 4 = -14 $
- $ -9 \times 8 = -72 $, $ -9 + 8 = -1 $
Wait: Try $ -12 \times 6 $? No.
Try $ -18 \times 4 $? No.
Try $ -12 \times 6 = -72 $, sum = -6 → no
Try $ -18 \times 4 = -72 $, sum = -14 → no
Try $ -9 \times 8 = -72 $, sum = -1 → no
Try $ -6 \times 12 = -72 $, sum = 6 → no
Try $ -4 \times 18 = -72 $, sum = 14 → no
Try $ -3 \times 24 = -72 $, sum = 21 → no
Try $ -2 \times 36 = -72 $, sum = 34 → no

Wait: $ -12 \times 6 = -72 $, but $ -12 + 6 = -6 $ → no
What about $ -18 \times 4 = -72 $, sum = -14 → no
Wait: $ -12 \times 6 = -72 $, but sum is -6
Wait: try $ -12 \times 6 = -72 $, no

Wait: $ -12 \times 6 = -72 $, sum = -6 → no

Wait: let's list all factor pairs of 72:
- $ 1,72 $ → sum 73
- $ 2,36 $ → 38
- $ 3,24 $ → 27
- $ 4,18 $ → 22
- $ 6,12 $ → 18
- $ 8,9 $ → 17

Now negative:
- $ -1,72 $ → 71
- $ -2,36 $ → 34
- $ -3,24 $ → 21
- $ -4,18 $ → 14
- $ -6,12 $ → 6
- $ -8,9 $ → 1
- $ -9,8 $ → -1
- $ -12,6 $ → -6
- $ -18,4 $ → -14
- $ -24,3 $ → -21
- $ -36,2 $ → -34
- $ -72,1 $ → -71

None sum to -8 → So no integer solution

> ✘ Prime (not factorable over integers)

Wait! Let me double-check:

Is there a pair that multiplies to -72, adds to -8?

Try $ -12 \times 6 = -72 $, sum = -6 → no
Try $ -18 \times 4 = -72 $, sum = -14 → no
Try $ -9 \times 8 = -72 $, sum = -1 → no
Try $ -6 \times 12 = -72 $, sum = 6 → no
Try $ -4 \times 18 = -72 $, sum = 14 → no
Try $ -3 \times 24 = -72 $, sum = 21 → no
Try $ -2 \times 36 = -72 $, sum = 34 → no
Try $ -1 \times 72 = -72 $, sum = 71 → no

Wait — what about $ -12 \times 6 $? No.

Wait: $ -18 \times 4 = -72 $, sum = -14 → no

Wait — perhaps I made a mistake.

Wait: $ -12 \times 6 = -72 $, sum = -6 → no

Wait: $ -8 \times 9 = -72 $, sum = 1 → no

Wait: $ -12 \times 6 = -72 $, sum = -6 → no

Wait: $ -18 \times 4 = -72 $, sum = -14 → no

Wait: $ -12 \times 6 = -72 $, sum = -6 → no

Actually, none work → so it's prime

> ✔ Answer: Prime

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19) $ y^2 - 4y - 60 $

Multiply to -60, add to -4:
- $ -10 \times 6 = -60 $, $ -10 + 6 = -4 $
$$
(y - 10)(y + 6)
$$

---

20) $ y^2 - 4y - 12 $

Multiply to -12, add to -4:
- $ -6 \times 2 = -12 $, $ -6 + 2 = -4 $
$$
(y - 6)(y + 2)
$$

---

21) $ u^2 + uv - 6v^2 $

Treat as quadratic in $ u $: $ u^2 + uv - 6v^2 $

Find two terms that multiply to $ -6v^2 $, add to $ v $:
- $ 3v \times -2v = -6v^2 $, $ 3v + (-2v) = v $
$$
(u + 3v)(u - 2v)
$$

---

22) $ x^2 + 4xy - 12y^2 $

Multiply to $ -12y^2 $, add to $ 4y $:
- $ 6y \times -2y = -12y^2 $, $ 6y + (-2y) = 4y $
$$
(x + 6y)(x - 2y)
$$

---

23) $ x^2 + 7xy - 30y^2 $

Multiply to $ -30y^2 $, add to $ 7y $:
- $ 10y \times -3y = -30y^2 $, $ 10y + (-3y) = 7y $
$$
(x + 10y)(x - 3y)
$$

---

24) $ x^2 + 13xy + 36y^2 $

Multiply to $ 36y^2 $, add to $ 13y $:
- $ 9y \times 4y = 36y^2 $, $ 9y + 4y = 13y $
$$
(x + 9y)(x + 4y)
$$

---

25) $ x^2 + 9xy + 18y^2 $

Multiply to $ 18y^2 $, add to $ 9y $:
- $ 6y \times 3y = 18y^2 $, $ 6y + 3y = 9y $
$$
(x + 6y)(x + 3y)
$$

---

26) $ x^2 - 12xy + 32y^2 $

Multiply to $ 32y^2 $, add to $ -12y $:
- $ -8y \times -4y = 32y^2 $, $ -8y + (-4y) = -12y $
$$
(x - 8y)(x - 4y)
$$

---

27) $ x^2 - 14xy + 48y^2 $

Multiply to $ 48y^2 $, add to $ -14y $:
- $ -8y \times -6y = 48y^2 $, $ -8y + (-6y) = -14y $
$$
(x - 8y)(x - 6y)
$$

---

28) $ x^2 + 3xy - 40y^2 $

Multiply to $ -40y^2 $, add to $ 3y $:
- $ 8y \times -5y = -40y^2 $, $ 8y + (-5y) = 3y $
$$
(x + 8y)(x - 5y)
$$

---

29) $ m^2 - 7mn - 8n^2 $

Multiply to $ -8n^2 $, add to $ -7n $:
- $ -8n \times n = -8n^2 $, $ -8n + n = -7n $
$$
(m - 8n)(m + n)
$$

---

30) $ x^2 - 8xy + 7y^2 $

Multiply to $ 7y^2 $, add to $ -8y $:
- $ -7y \times -y = 7y^2 $, $ -7y + (-y) = -8y $
$$
(x - 7y)(x - y)
$$

---

31) $ x^2 + 2xy + 4y^2 $

Check if perfect square:
- $ (x + 2y)^2 = x^2 + 4xy + 4y^2 $ → not same
- Middle term is $ 2xy $, but should be $ 4xy $
→ Not a perfect square

Try factoring:
Multiply to $ 4y^2 $, add to $ 2y $:
- $ 2y \times 2y = 4y^2 $, $ 2y + 2y = 4y $ → too big
- $ 4y \times y = 4y^2 $, $ 4y + y = 5y $
- $ -2y \times -2y = 4y^2 $, $ -2y + (-2y) = -4y $
- $ 1y \times 4y = 4y^2 $, sum = 5y
No pair sums to $ 2y $

> ✘ Prime

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32) $ a^2 + 4ab - 5b^2 $

Multiply to $ -5b^2 $, add to $ 4b $:
- $ 5b \times -b = -5b^2 $, $ 5b + (-b) = 4b $
$$
(a + 5b)(a - b)
$$

---

33) $ x^2 + 12xy + 32y^2 $

Multiply to $ 32y^2 $, add to $ 12y $:
- $ 8y \times 4y = 32y^2 $, $ 8y + 4y = 12y $
$$
(x + 8y)(x + 4y)
$$

---

34) $ x^2 + 9xy + 14y^2 $

Multiply to $ 14y^2 $, add to $ 9y $:
- $ 7y \times 2y = 14y^2 $, $ 7y + 2y = 9y $
$$
(x + 7y)(x + 2y)
$$

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35) $ u^2 - 12uv + 20v^2 $

Multiply to $ 20v^2 $, add to $ -12v $:
- $ -10v \times -2v = 20v^2 $, $ -10v + (-2v) = -12v $
$$
(u - 10v)(u - 2v)
$$

---

36) $ u^2 - uv - 2v^2 $

Multiply to $ -2v^2 $, add to $ -v $:
- $ -2v \times v = -2v^2 $, $ -2v + v = -v $
$$
(u - 2v)(u + v)
$$

---

37) $ x^2 + 2xy + 12y^2 $

Multiply to $ 12y^2 $, add to $ 2y $:
- $ 6y \times 2y = 12y^2 $, sum = 8y
- $ 4y \times 3y = 12y^2 $, sum = 7y
- $ 12y \times y = 12y^2 $, sum = 13y
- $ -6y \times -2y = 12y^2 $, sum = -8y
No pair sums to $ 2y $

> ✘ Prime

---

38) $ m^2 + 13mn + 40n^2 $

Multiply to $ 40n^2 $, add to $ 13n $:
- $ 8n \times 5n = 40n^2 $, $ 8n + 5n = 13n $
$$
(m + 8n)(m + 5n)
$$

---

39) $ u^2 + 8uv + 12v^2 $

Multiply to $ 12v^2 $, add to $ 8v $:
- $ 6v \times 2v = 12v^2 $, $ 6v + 2v = 8v $
$$
(u + 6v)(u + 2v)
$$

---

40) $ x^2 + 10xy + 16y^2 $

Multiply to $ 16y^2 $, add to $ 10y $:
- $ 8y \times 2y = 16y^2 $, $ 8y + 2y = 10y $
$$
(x + 8y)(x + 2y)
$$

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## ✔ Final Answers:

| Problem | Factored Form |
|--------|----------------|
| 1) | $ (x + 6)(x + 1) $ |
| 2) | $ (x - 6)(x - 1) $ |
| 3) | $ (p - 9)(p + 5) $ |
| 4) | $ (b - 6)(b - 2) $ |
| 5) | $ (x - 8)(x + 1) $ |
| 6) | Prime |
| 7) | $ (a + 6)(a + 8) $ |
| 8) | $ (n - 10)(n - 5) $ |
| 9) | $ (p + 2)^2 $ |
| 10) | $ (x - 9)(x + 3) $ |
| 11) | Prime |
| 12) | $ (x + 9)(x - 8) $ |
| 13) | $ (x - 9)(x + 1) $ |
| 14) | $ (p + 6)^2 $ |
| 15) | $ (n - 9)(n + 2) $ |
| 16) | $ (m + 7)(m + 4) $ |
| 17) | $ (m + 9)(m + 3) $ |
| 18) | Prime |
| 19) | $ (y - 10)(y + 6) $ |
| 20) | $ (y - 6)(y + 2) $ |
| 21) | $ (u + 3v)(u - 2v) $ |
| 22) | $ (x + 6y)(x - 2y) $ |
| 23) | $ (x + 10y)(x - 3y) $ |
| 24) | $ (x + 9y)(x + 4y) $ |
| 25) | $ (x + 6y)(x + 3y) $ |
| 26) | $ (x - 8y)(x - 4y) $ |
| 27) | $ (x - 8y)(x - 6y) $ |
| 28) | $ (x + 8y)(x - 5y) $ |
| 29) | $ (m - 8n)(m + n) $ |
| 30) | $ (x - 7y)(x - y) $ |
| 31) | Prime |
| 32) | $ (a + 5b)(a - b) $ |
| 33) | $ (x + 8y)(x + 4y) $ |
| 34) | $ (x + 7y)(x + 2y) $ |
| 35) | $ (u - 10v)(u - 2v) $ |
| 36) | $ (u - 2v)(u + v) $ |
| 37) | Prime |
| 38) | $ (m + 8n)(m + 5n) $ |
| 39) | $ (u + 6v)(u + 2v) $ |
| 40) | $ (x + 8y)(x + 2y) $ |

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🔍 Summary of Strategy:

To factor $ x^2 + bx + c $:
1. Find two numbers that:
- Multiply to $ c $
- Add to $ b $
2. Write as $ (x + m)(x + n) $
3. If no such integers exist → Prime

For variables like $ xy $, treat one variable as constant and find combinations accordingly.

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