ACTIVITY MAZE INB - Algebra - Factoring Quadratics - a > 1 ... - Free Printable

To factor each quadratic expression completely, we will use the method of recognizing perfect square trinomials. A perfect square trinomial is of the form:

1. \( (a + b)^2 = a^2 + 2ab + b^2 \)
2. \( (a - b)^2 = a^2 - 2ab + b^2 \)

Let's factor each expression step by step.

---

1. \( 25s^2 + 30s + 9 \)

- Identify the structure: This looks like a perfect square trinomial.
- Compare with \( a^2 + 2ab + b^2 \):
- \( a^2 = 25s^2 \implies a = 5s \)
- \( b^2 = 9 \implies b = 3 \)
- \( 2ab = 2(5s)(3) = 30s \), which matches the middle term.
- Factor:
\[
25s^2 + 30s + 9 = (5s + 3)^2
\]

---

2. \( 9g^2 - 42g + 49 \)

- Identify the structure: This looks like a perfect square trinomial.
- Compare with \( a^2 - 2ab + b^2 \):
- \( a^2 = 9g^2 \implies a = 3g \)
- \( b^2 = 49 \implies b = 7 \)
- \( -2ab = -2(3g)(7) = -42g \), which matches the middle term.
- Factor:
\[
9g^2 - 42g + 49 = (3g - 7)^2
\]

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3. \( 16r^2 + 56r + 49 \)

- Identify the structure: This looks like a perfect square trinomial.
- Compare with \( a^2 + 2ab + b^2 \):
- \( a^2 = 16r^2 \implies a = 4r \)
- \( b^2 = 49 \implies b = 7 \)
- \( 2ab = 2(4r)(7) = 56r \), which matches the middle term.
- Factor:
\[
16r^2 + 56r + 49 = (4r + 7)^2
\]

---

4. \( 4p^2 + 36p + 81 \)

- Identify the structure: This looks like a perfect square trinomial.
- Compare with \( a^2 + 2ab + b^2 \):
- \( a^2 = 4p^2 \implies a = 2p \)
- \( b^2 = 81 \implies b = 9 \)
- \( 2ab = 2(2p)(9) = 36p \), which matches the middle term.
- Factor:
\[
4p^2 + 36p + 81 = (2p + 9)^2
\]

---

5. \( z^2 - 2z + 1 \)

- Identify the structure: This looks like a perfect square trinomial.
- Compare with \( a^2 - 2ab + b^2 \):
- \( a^2 = z^2 \implies a = z \)
- \( b^2 = 1 \implies b = 1 \)
- \( -2ab = -2(z)(1) = -2z \), which matches the middle term.
- Factor:
\[
z^2 - 2z + 1 = (z - 1)^2
\]

---

6. \( 16g^2 - 24g + 9 \)

- Identify the structure: This looks like a perfect square trinomial.
- Compare with \( a^2 - 2ab + b^2 \):
- \( a^2 = 16g^2 \implies a = 4g \)
- \( b^2 = 9 \implies b = 3 \)
- \( -2ab = -2(4g)(3) = -24g \), which matches the middle term.
- Factor:
\[
16g^2 - 24g + 9 = (4g - 3)^2
\]

---

7. \( 9k^2 - 24k + 16 \)

- Identify the structure: This looks like a perfect square trinomial.
- Compare with \( a^2 - 2ab + b^2 \):
- \( a^2 = 9k^2 \implies a = 3k \)
- \( b^2 = 16 \implies b = 4 \)
- \( -2ab = -2(3k)(4) = -24k \), which matches the middle term.
- Factor:
\[
9k^2 - 24k + 16 = (3k - 4)^2
\]

---

8. \( 4s^2 + 4s + 1 \)

- Identify the structure: This looks like a perfect square trinomial.
- Compare with \( a^2 + 2ab + b^2 \):
- \( a^2 = 4s^2 \implies a = 2s \)
- \( b^2 = 1 \implies b = 1 \)
- \( 2ab = 2(2s)(1) = 4s \), which matches the middle term.
- Factor:
\[
4s^2 + 4s + 1 = (2s + 1)^2
\]

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Final Answer

\[
\boxed{
\begin{aligned}
&25s^2 + 30s + 9 = (5s + 3)^2 \\
&9g^2 - 42g + 49 = (3g - 7)^2 \\
&16r^2 + 56r + 49 = (4r + 7)^2 \\
&4p^2 + 36p + 81 = (2p + 9)^2 \\
&z^2 - 2z + 1 = (z - 1)^2 \\
&16g^2 - 24g + 9 = (4g - 3)^2 \\
&9k^2 - 24k + 16 = (3k - 4)^2 \\
&4s^2 + 4s + 1 = (2s + 1)^2
\end{aligned}
}
\]