Math worksheet featuring algebraic expressions to expand and simplify, including binomial multiplication and squaring.

Problem Analysis:

The task involves expanding and simplifying algebraic expressions. We will solve each part step by step, ensuring clarity in the process.

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Task 3: Expand and simplify the following brackets

#### (a) \( (x + 3)(x + 4) \)
Using the distributive property (also known as the FOIL method for binomials):
\[
(x + 3)(x + 4) = x(x) + x(4) + 3(x) + 3(4)
\]
\[
= x^2 + 4x + 3x + 12
\]
\[
= x^2 + 7x + 12
\]

#### (b) \( (x + 5)(x + 2) \)
Similarly:
\[
(x + 5)(x + 2) = x(x) + x(2) + 5(x) + 5(2)
\]
\[
= x^2 + 2x + 5x + 10
\]
\[
= x^2 + 7x + 10
\]

#### (c) \( (y + 6)(y + 3) \)
\[
(y + 6)(y + 3) = y(y) + y(3) + 6(y) + 6(3)
\]
\[
= y^2 + 3y + 6y + 18
\]
\[
= y^2 + 9y + 18
\]

#### (d) \( (h + 8)(h + 5) \)
\[
(h + 8)(h + 5) = h(h) + h(5) + 8(h) + 8(5)
\]
\[
= h^2 + 5h + 8h + 40
\]
\[
= h^2 + 13h + 40
\]

#### (e) \( (f + 9)(f - 2) \)
\[
(f + 9)(f - 2) = f(f) + f(-2) + 9(f) + 9(-2)
\]
\[
= f^2 - 2f + 9f - 18
\]
\[
= f^2 + 7f - 18
\]

#### (f) \( (g + 8)(g - 3) \)
\[
(g + 8)(g - 3) = g(g) + g(-3) + 8(g) + 8(-3)
\]
\[
= g^2 - 3g + 8g - 24
\]
\[
= g^2 + 5g - 24
\]

#### (g) \( (r - 6)(r + 4) \)
\[
(r - 6)(r + 4) = r(r) + r(4) - 6(r) - 6(4)
\]
\[
= r^2 + 4r - 6r - 24
\]
\[
= r^2 - 2r - 24
\]

#### (h) \( (p - 7)(p - 5) \)
\[
(p - 7)(p - 5) = p(p) + p(-5) - 7(p) - 7(-5)
\]
\[
= p^2 - 5p - 7p + 35
\]
\[
= p^2 - 12p + 35
\]

#### (i) \( (n - 9)(n - 4) \)
\[
(n - 9)(n - 4) = n(n) + n(-4) - 9(n) - 9(-4)
\]
\[
= n^2 - 4n - 9n + 36
\]
\[
= n^2 - 13n + 36
\]

#### (j) \( (d - 8)(d + 4) \)
\[
(d - 8)(d + 4) = d(d) + d(4) - 8(d) - 8(4)
\]
\[
= d^2 + 4d - 8d - 32
\]
\[
= d^2 - 4d - 32
\]

---

Task 4: Expand these brackets and write down any patterns you notice

#### (a) \( (x + 3)(x - 3) \)
This is a difference of squares:
\[
(x + 3)(x - 3) = x^2 - 3^2
\]
\[
= x^2 - 9
\]

#### (b) \( (x - 5)(x + 5) \)
Again, a difference of squares:
\[
(x - 5)(x + 5) = x^2 - 5^2
\]
\[
= x^2 - 25
\]

#### (c) \( (y + 6)(y - 6) \)
Difference of squares:
\[
(y + 6)(y - 6) = y^2 - 6^2
\]
\[
= y^2 - 36
\]

#### (d) \( (h + 8)(h - 8) \)
Difference of squares:
\[
(h + 8)(h - 8) = h^2 - 8^2
\]
\[
= h^2 - 64
\]

#### (e) \( (f - 9)(f + 9) \)
Difference of squares:
\[
(f - 9)(f + 9) = f^2 - 9^2
\]
\[
= f^2 - 81
\]

#### (f) \( (g + 4)(g - 4) \)
Difference of squares:
\[
(g + 4)(g - 4) = g^2 - 4^2
\]
\[
= g^2 - 16
\]

Pattern Noticed: All these expansions result in the form \( a^2 - b^2 \), which is the difference of squares formula:
\[
(a + b)(a - b) = a^2 - b^2
\]

---

Task 5: Expand and simplify the following brackets

#### (a) \( (x + 3)^2 \)
Using the square of a binomial formula \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[
(x + 3)^2 = x^2 + 2(x)(3) + 3^2
\]
\[
= x^2 + 6x + 9
\]

#### (b) \( (x - 5)^2 \)
Using the square of a binomial formula \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[
(x - 5)^2 = x^2 - 2(x)(5) + 5^2
\]
\[
= x^2 - 10x + 25
\]

#### (c) \( (y + 6)^2 \)
\[
(y + 6)^2 = y^2 + 2(y)(6) + 6^2
\]
\[
= y^2 + 12y + 36
\]

#### (d) \( (h + 8)^2 \)
\[
(h + 8)^2 = h^2 + 2(h)(8) + 8^2
\]
\[
= h^2 + 16h + 64
\]

#### (e) \( (f + 9)^2 \)
\[
(f + 9)^2 = f^2 + 2(f)(9) + 9^2
\]
\[
= f^2 + 18f + 81
\]

#### (f) \( (g - 8)^2 \)
\[
(g - 8)^2 = g^2 - 2(g)(8) + 8^2
\]
\[
= g^2 - 16g + 64
\]

#### (g) \( (r - 6)^2 \)
\[
(r - 6)^2 = r^2 - 2(r)(6) + 6^2
\]
\[
= r^2 - 12r + 36
\]

#### (h) \( (p + 7)^2 \)
\[
(p + 7)^2 = p^2 + 2(p)(7) + 7^2
\]
\[
= p^2 + 14p + 49
\]

#### (i) \( (n - 9)^2 \)
\[
(n - 9)^2 = n^2 - 2(n)(9) + 9^2
\]
\[
= n^2 - 18n + 81
\]

#### (j) \( (d - 2)^2 \)
\[
(d - 2)^2 = d^2 - 2(d)(2) + 2^2
\]
\[
= d^2 - 4d + 4
\]

---

Final Answers:

Task 3:
\[
\boxed{
\begin{aligned}
&\text{(a)}\ x^2 + 7x + 12, &\text{(b)}\ x^2 + 7x + 10, &\text{(c)}\ y^2 + 9y + 18, \\
&\text{(d)}\ h^2 + 13h + 40, &\text{(e)}\ f^2 + 7f - 18, &\text{(f)}\ g^2 + 5g - 24, \\
&\text{(g)}\ r^2 - 2r - 24, &\text{(h)}\ p^2 - 12p + 35, &\text{(i)}\ n^2 - 13n + 36, \\
&\text{(j)}\ d^2 - 4d - 32.
\end{aligned}
}
\]

Task 4:
\[
\boxed{
\begin{aligned}
&\text{(a)}\ x^2 - 9, &\text{(b)}\ x^2 - 25, &\text{(c)}\ y^2 - 36, \\
&\text{(d)}\ h^2 - 64, &\text{(e)}\ f^2 - 81, &\text{(f)}\ g^2 - 16.
\end{aligned}
}
\]

Task 5:
\[
\boxed{
\begin{aligned}
&\text{(a)}\ x^2 + 6x + 9, &\text{(b)}\ x^2 - 10x + 25, &\text{(c)}\ y^2 + 12y + 36, \\
&\text{(d)}\ h^2 + 16h + 64, &\text{(e)}\ f^2 + 18f + 81, &\text{(f)}\ g^2 - 16g + 64, \\
&\text{(g)}\ r^2 - 12r + 36, &\text{(h)}\ p^2 + 14p + 49, &\text{(i)}\ n^2 - 18n + 81, \\
&\text{(j)}\ d^2 - 4d + 4.
\end{aligned}
}
\]