Lesson plan in mathematics 8 (Factoring Perfect Square Trinomial ... - Free Printable

Problem Description:

The task involves solving a series of algebraic expressions involving binomials. The expressions are given in the form of puzzles, and students are expected to solve them by expanding the binomials and simplifying the resulting expressions.

Expressions to Solve:

1. \( (x + 3)^2 \)
2. \( (x - 7)^2 \)
3. \( (x + 2)(x + 7) \)
4. \( (x - 4)(x + 5) \)
5. \( (x - 6)(x + 2) \)

Solution:

#### 1. \( (x + 3)^2 \)

Using the formula for the square of a binomial:
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
Here, \( a = x \) and \( b = 3 \).

\[
(x + 3)^2 = x^2 + 2(x)(3) + 3^2
\]
\[
= x^2 + 6x + 9
\]

Answer: \( x^2 + 6x + 9 \)

---

#### 2. \( (x - 7)^2 \)

Using the same formula for the square of a binomial:
\[
(a - b)^2 = a^2 - 2ab + b^2
\]
Here, \( a = x \) and \( b = 7 \).

\[
(x - 7)^2 = x^2 - 2(x)(7) + 7^2
\]
\[
= x^2 - 14x + 49
\]

Answer: \( x^2 - 14x + 49 \)

---

#### 3. \( (x + 2)(x + 7) \)

Using the distributive property (FOIL method):
\[
(a + b)(c + d) = ac + ad + bc + bd
\]
Here, \( a = x \), \( b = 2 \), \( c = x \), and \( d = 7 \).

\[
(x + 2)(x + 7) = x(x) + x(7) + 2(x) + 2(7)
\]
\[
= x^2 + 7x + 2x + 14
\]
\[
= x^2 + 9x + 14
\]

Answer: \( x^2 + 9x + 14 \)

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#### 4. \( (x - 4)(x + 5) \)

Using the distributive property (FOIL method):
\[
(a - b)(c + d) = ac + ad - bc - bd
\]
Here, \( a = x \), \( b = 4 \), \( c = x \), and \( d = 5 \).

\[
(x - 4)(x + 5) = x(x) + x(5) - 4(x) - 4(5)
\]
\[
= x^2 + 5x - 4x - 20
\]
\[
= x^2 + x - 20
\]

Answer: \( x^2 + x - 20 \)

---

#### 5. \( (x - 6)(x + 2) \)

Using the distributive property (FOIL method):
\[
(a - b)(c + d) = ac + ad - bc - bd
\]
Here, \( a = x \), \( b = 6 \), \( c = x \), and \( d = 2 \).

\[
(x - 6)(x + 2) = x(x) + x(2) - 6(x) - 6(2)
\]
\[
= x^2 + 2x - 6x - 12
\]
\[
= x^2 - 4x - 12
\]

Answer: \( x^2 - 4x - 12 \)

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Final Answers:

1. \( (x + 3)^2 = \boxed{x^2 + 6x + 9} \)
2. \( (x - 7)^2 = \boxed{x^2 - 14x + 49} \)
3. \( (x + 2)(x + 7) = \boxed{x^2 + 9x + 14} \)
4. \( (x - 4)(x + 5) = \boxed{x^2 + x - 20} \)
5. \( (x - 6)(x + 2) = \boxed{x^2 - 4x - 12} \)

Explanation:

- For squaring binomials (\( (a + b)^2 \) or \( (a - b)^2 \)), we use the formulas \( a^2 + 2ab + b^2 \) and \( a^2 - 2ab + b^2 \), respectively.
- For multiplying two binomials, we use the distributive property (FOIL method) to expand and simplify the expression.
- Each step involves careful application of algebraic rules to ensure accuracy.