Factor quadratics that are perfect squares with this printable math worksheet.

Let's solve each quadratic expression by factoring perfect square trinomials. A perfect square trinomial has the form:

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

We'll go through each one step by step.

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1. $ 25s^2 + 30s + 9 $

- First term: $ 25s^2 = (5s)^2 $
- Last term: $ 9 = 3^2 $
- Middle term: $ 30s = 2 \cdot (5s) \cdot 3 $

So this fits the pattern:
$$
(5s)^2 + 2(5s)(3) + 3^2 = (5s + 3)^2
$$

✔ Answer: $ (5s + 3)^2 $

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2. $ 9g^2 - 42g + 49 $

- First term: $ 9g^2 = (3g)^2 $
- Last term: $ 49 = 7^2 $
- Middle term: $ -42g = -2 \cdot (3g) \cdot 7 $

So:
$$
(3g)^2 - 2(3g)(7) + 7^2 = (3g - 7)^2
$$

✔ Answer: $ (3g - 7)^2 $

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3. $ 16r^2 + 56r + 49 $

- First term: $ 16r^2 = (4r)^2 $
- Last term: $ 49 = 7^2 $
- Middle term: $ 56r = 2 \cdot (4r) \cdot 7 $

So:
$$
(4r)^2 + 2(4r)(7) + 7^2 = (4r + 7)^2
$$

✔ Answer: $ (4r + 7)^2 $

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4. $ 4p^2 + 36p + 81 $

- First term: $ 4p^2 = (2p)^2 $
- Last term: $ 81 = 9^2 $
- Middle term: $ 36p = 2 \cdot (2p) \cdot 9 $

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

✔ Answer: $ (2p + 9)^2 $

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5. $ z^2 - 2z + 1 $

- First term: $ z^2 = (z)^2 $
- Last term: $ 1 = 1^2 $
- Middle term: $ -2z = -2 \cdot z \cdot 1 $

So:
$$
z^2 - 2z + 1 = (z - 1)^2
$$

✔ Answer: $ (z - 1)^2 $

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6. $ 16g^2 - 24g + 9 $

- First term: $ 16g^2 = (4g)^2 $
- Last term: $ 9 = 3^2 $
- Middle term: $ -24g = -2 \cdot (4g) \cdot 3 $

So:
$$
(4g)^2 - 2(4g)(3) + 3^2 = (4g - 3)^2
$$

✔ Answer: $ (4g - 3)^2 $

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7. $ 9k^2 - 24k + 16 $

- First term: $ 9k^2 = (3k)^2 $
- Last term: $ 16 = 4^2 $
- Middle term: $ -24k = -2 \cdot (3k) \cdot 4 $

So:
$$
(3k)^2 - 2(3k)(4) + 4^2 = (3k - 4)^2
$$

✔ Answer: $ (3k - 4)^2 $

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8. $ 4s^2 + 4s + 1 $

- First term: $ 4s^2 = (2s)^2 $
- Last term: $ 1 = 1^2 $
- Middle term: $ 4s = 2 \cdot (2s) \cdot 1 $

So:
$$
(2s)^2 + 2(2s)(1) + 1^2 = (2s + 1)^2
$$

✔ Answer: $ (2s + 1)^2 $

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✔ Final Answers:

| Expression | Factored Form |
|----------|----------------|
| $ 25s^2 + 30s + 9 $ | $ (5s + 3)^2 $ |
| $ 9g^2 - 42g + 49 $ | $ (3g - 7)^2 $ |
| $ 16r^2 + 56r + 49 $ | $ (4r + 7)^2 $ |
| $ 4p^2 + 36p + 81 $ | $ (2p + 9)^2 $ |
| $ z^2 - 2z + 1 $ | $ (z - 1)^2 $ |
| $ 16g^2 - 24g + 9 $ | $ (4g - 3)^2 $ |
| $ 9k^2 - 24k + 16 $ | $ (3k - 4)^2 $ |
| $ 4s^2 + 4s + 1 $ | $ (2s + 1)^2 $ |

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💡 Tip: Always check if the first and last terms are perfect squares, then verify if the middle term is twice the product of their square roots. If yes, it's a perfect square trinomial!

Let me know if you'd like a visual explanation or practice problems!