📐 Concept diagram

01-07 - Quadratic Expressions

Phase: 1 | Subject: 01-07 Prerequisites: 01-06-systems-of-linear-equations.md (algebraic manipulation) Next subject: 01-08-quadratic-equations.md

Learning Objectives

By the end of this subject, you will be able to:

1. Expand perfect square expressions: (a+b)² and (a-b)²
2. Expand the difference of two squares: (a+b)(a-b)
3. Factorise quadratic expressions of the form x²+bx+c
4. Factorise quadratic expressions of the form ax²+bx+c
5. Factorise using the difference of two squares pattern

Core Content

Special Binomial Products

(a + b)² — Perfect Square

$(a + b)² = a² + 2ab + b²$

Mnemonic: "Square the first, double the product, square the last"

Common pitfall: (a + b)² is NOT a² + b². The middle term 2ab is essential. A geometric way to see this: draw a square of side (a+b). Its area is a² + ab + ab + b² = a² + 2ab + b².

Example: (x + 4)² = x² + 2(x)(4) + 16 = x² + 8x + 16

Example: (2y + 3)² = 4y² + 2(2y)(3) + 9 = 4y² + 12y + 9

(a - b)² — Perfect Square

$(a - b)² = a² - 2ab + b²$

Example: (x - 5)² = x² - 2(x)(5) + 25 = x² - 10x + 25

Example: (3y - 2)² = 9y² - 2(3y)(2) + 4 = 9y² - 12y + 4

(a + b)(a - b) — Difference of Two Squares

$(a + b)(a - b) = a² - b²$

The middle terms cancel: +ab and -ab.

Example: (x + 7)(x - 7) = x² - 49

Example: (3y + 2)(3y - 2) = 9y² - 4

Factorising Quadratics: x² + bx + c

Find two numbers that: - MULTIPLY to c - ADD to b

Example: Factorise x² + 7x + 12

1. Find numbers that multiply to 12 and add to 7: 3 and 4
2. Check: 3 × 4 = 12 ✓, 3 + 4 = 7 ✓
3. Result: (x + 3)(x + 4)

Verify: (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12 ✓

Example: Factorise x² - 5x + 6

1. Find numbers that multiply to 6 and add to -5: -2 and -3
2. Result: (x - 2)(x - 3)

Example: Factorise x² + x - 12

1. Find numbers that multiply to -12 and add to 1: 4 and -3
2. Result: (x + 4)(x - 3)

Factorising Quadratics: ax² + bx + c

Method: Splitting the middle term

Example: Factorise 2x² + 7x + 3

1. Multiply a × c: 2 × 3 = 6
2. Find two numbers that multiply to 6 and add to 7: 6 and 1
3. Split the middle term: 2x² + 6x + x + 3
4. Factor in pairs:
5. 2x² + 6x = 2x(x + 3)
6. x + 3 = 1(x + 3)
7. Common factor: (x + 3)
8. Result: (2x + 1)(x + 3)

Verify: (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3 ✓

Example: Factorise 3x² - 10x + 8

1. a × c = 3 × 8 = 24
2. Find numbers that multiply to 24 and add to -10: -4 and -6
3. Split: 3x² - 4x - 6x + 8
4. Factor in pairs:
5. 3x² - 4x = x(3x - 4)
6. -6x + 8 = -2(3x - 4)
7. Common factor: (3x - 4)
8. Result: (3x - 4)(x - 2) or (x - 2)(3x - 4)

Factorising Using Difference of Two Squares

Example: Factorise 4x² - 25

1. Both terms are perfect squares: 4x² = (2x)², 25 = 5²
2. Result: (2x + 5)(2x - 5)

Example: Factorise x⁴ - 16

1. x⁴ = (x²)², 16 = 4²
2. First factorisation: (x² + 4)(x² - 4)
3. But x² - 4 is itself a difference of squares: (x + 2)(x - 2)
4. Complete factorisation: (x² + 4)(x + 2)(x - 2)

Key Terms

• (a + b)(a - b) — Difference of Two Squares
• (a + b)² — Perfect Square
• (a - b)² — Perfect Square
• 01 07 Quadratic Expressions
• Correct: A)
• Correct: B)
• Example 1: Expand and simplify (2x - 3)²
• Example 2: Factorise x² + 10x + 24
• Example 3: Factorise 6x² - 5x - 4
• Factorising Quadratics: ax² + bx + c
• Factorising Quadratics: x² + bx + c
• Factorising Using Difference of Two Squares

Worked Examples

Example 1: Expand and simplify (2x - 3)²

1. Use formula: (a - b)² = a² - 2ab + b²
2. a = 2x, b = 3
3. (2x)² - 2(2x)(3) + 9 = 4x² - 12x + 9

Example 2: Factorise x² + 10x + 24

1. Find numbers that multiply to 24 and add to 10: 4 and 6
2. Result: (x + 4)(x + 6)

Example 3: Factorise 6x² - 5x - 4

1. a × c = 6 × (-4) = -24
2. Find numbers that multiply to -24 and add to -5: -8 and 3
3. Split: 6x² - 8x + 3x - 4
4. Factor in pairs:
5. 6x² - 8x = 2x(3x - 4)
6. 3x - 4 = 1(3x - 4)
7. Result: (2x + 1)(3x - 4)

Quiz

Q1: What does the concept of Factorising Using Difference of Two Squares primarily refer to in this subject?

A) A visual representation of Factorising Using Difference of Two Squares B) A computational error related to Factorising Using Difference of Two Squares C) The definition and application of Factorising Using Difference of Two Squares D) A historical anecdote about Factorising Using Difference of Two Squares

Correct: C)

• If you chose A: This is incorrect. Factorising Using Difference of Two Squares is defined as: the definition and application of factorising using difference of two squares. The other options describe different aspects that are not the primary focus.
• If you chose B: This is incorrect. Factorising Using Difference of Two Squares is defined as: the definition and application of factorising using difference of two squares. The other options describe different aspects that are not the primary focus.
• If you chose C: Factorising Using Difference of Two Squares is defined as: the definition and application of factorising using difference of two squares. The other options describe different aspects that are not the primary focus. Correct!
• If you chose D: This is incorrect. Factorising Using Difference of Two Squares is defined as: the definition and application of factorising using difference of two squares. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of Special Binomial Products?

A) It is primarily a historical notation system B) It is used to special binomial products in mathematical analysis C) It is used only in advanced research contexts D) It replaces all other methods in this domain

Correct: B)

• If you chose A: This is incorrect. Special Binomial Products serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: Special Binomial Products serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose C: This is incorrect. Special Binomial Products serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose D: This is incorrect. Special Binomial Products serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about (A + B)² — Perfect Square is TRUE?

A) (A + B)² — Perfect Square is not related to this subject B) (A + B)² — Perfect Square is mentioned only as a historical footnote C) (A + B)² — Perfect Square is an advanced topic beyond this subject's scope D) (A + B)² — Perfect Square is a fundamental concept covered in this subject

Correct: D)

• If you chose A: This is incorrect. (A + B)² — Perfect Square is a fundamental concept covered in this subject. This subject covers (A + B)² — Perfect Square as part of its core content.
• If you chose B: This is incorrect. (A + B)² — Perfect Square is a fundamental concept covered in this subject. This subject covers (A + B)² — Perfect Square as part of its core content.
• If you chose C: This is incorrect. (A + B)² — Perfect Square is a fundamental concept covered in this subject. This subject covers (A + B)² — Perfect Square as part of its core content.
• If you chose D: (A + B)² — Perfect Square is a fundamental concept covered in this subject. This subject covers (A + B)² — Perfect Square as part of its core content. Correct!

Q4: Based on the worked examples in this subject, what is the correct result?

A) A different result from a common mistake B) The inverse of the correct answer C) (x + 3)(x + 4) D) An unrelated numerical value

Correct: C)

• If you chose A: This is incorrect. The worked examples show that the result is (x + 3)(x + 4). The other options represent common errors.
• If you chose B: This is incorrect. The worked examples show that the result is (x + 3)(x + 4). The other options represent common errors.
• If you chose C: The worked examples show that the result is (x + 3)(x + 4). The other options represent common errors. Correct!
• If you chose D: This is incorrect. The worked examples show that the result is (x + 3)(x + 4). The other options represent common errors.

Q5: How are (A + B)² — Perfect Square and (A - B)² — Perfect Square related?

A) (A + B)² — Perfect Square and (A - B)² — Perfect Square are closely related concepts B) (A + B)² — Perfect Square is the inverse of (A - B)² — Perfect Square C) (A + B)² — Perfect Square is a special case of (A - B)² — Perfect Square D) (A + B)² — Perfect Square and (A - B)² — Perfect Square are completely unrelated topics

Correct: A)

• If you chose A: Both (A + B)² — Perfect Square and (A - B)² — Perfect Square are covered in this subject as interconnected topics. Correct!
• If you chose B: This is incorrect. Both (A + B)² — Perfect Square and (A - B)² — Perfect Square are covered in this subject as interconnected topics.
• If you chose C: This is incorrect. Both (A + B)² — Perfect Square and (A - B)² — Perfect Square are covered in this subject as interconnected topics.
• If you chose D: This is incorrect. Both (A + B)² — Perfect Square and (A - B)² — Perfect Square are covered in this subject as interconnected topics.

Q6: What is a common pitfall when working with (A + B)(A - B) — Difference Of Two Squares?

A) A common mistake is confusing (A + B)(A - B) — Difference Of Two Squares with a similar concept B) The main error with (A + B)(A - B) — Difference Of Two Squares is using it when it is not needed C) (A + B)(A - B) — Difference Of Two Squares has no common misconceptions D) (A + B)(A - B) — Difference Of Two Squares is always computed the same way in all contexts

Correct: A)

• If you chose A: Students often confuse (A + B)(A - B) — Difference Of Two Squares with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose B: This is incorrect. Students often confuse (A + B)(A - B) — Difference Of Two Squares with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose C: This is incorrect. Students often confuse (A + B)(A - B) — Difference Of Two Squares with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: This is incorrect. Students often confuse (A + B)(A - B) — Difference Of Two Squares with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply Factorising Quadratics: X² + Bx + C?

A) Factorising Quadratics: X² + Bx + C is not practically useful B) Use Factorising Quadratics: X² + Bx + C only in pure mathematics contexts C) Apply Factorising Quadratics: X² + Bx + C to solve problems in this subject's domain D) Avoid Factorising Quadratics: X² + Bx + C unless explicitly instructed

Correct: C)

• If you chose A: This is incorrect. Factorising Quadratics: X² + Bx + C is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: This is incorrect. Factorising Quadratics: X² + Bx + C is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: Factorising Quadratics: X² + Bx + C is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose D: This is incorrect. Factorising Quadratics: X² + Bx + C is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. Expand: (x + 5)²
   Click for answer

x² + 10x + 25

1. Expand: (3y - 2)²
   Click for answer

9y² - 12y + 4

1. Expand: (2a + 7)(2a - 7)
   Click for answer

4a² - 49

1. Factorise: x² + 8x + 15
   Click for answer

(x + 3)(x + 5)

1. Factorise: x² - 7x + 12
   Click for answer

(x - 3)(x - 4)

1. Factorise: x² + 2x - 15
   Click for answer

(x + 5)(x - 3)

1. Factorise: 2x² + 9x + 4
   Click for answer

Split 9 into 8 + 1: 2x² + 8x + x + 4 = 2x(x + 4) + (x + 4) = (2x + 1)(x + 4)

1. Factorise: 3x² - 2x - 8
   Click for answer

a × c = -24. Split -2 into -6 + 4: 3x² - 6x + 4x - 8 = 3x(x - 2) + 4(x - 2) = (3x + 4)(x - 2)

1. Factorise: 9x² - 16
   Click for answer

(3x + 4)(3x - 4)

Summary

Key takeaways:

• (a + b)² = a² + 2ab + b² — "square first, double product, square last"
• (a - b)² = a² - 2ab + b² — same pattern, middle term is negative
• (a + b)(a - b) = a² - b² — difference of two squares
• To factorise x² + bx + c: find two numbers that multiply to c and add to b
• For ax² + bx + c: split middle term using numbers that multiply to ac and add to b
• Always verify by expanding back

Pitfalls

• Forgetting the middle term when expanding perfect squares. (a + b)² = a² + 2ab + b², NOT a² + b². The 2ab term is essential. A geometric visualisation — a square of side (a+b) divided into four regions — confirms that the middle term must appear.
• Confusing the difference of squares with a perfect square. (x + 5)(x - 5) = x² - 25 (difference of squares, no middle term). But (x - 5)² = x² - 10x + 25 (perfect square, HAS a middle term). These are different patterns — don't mix them up.
• Sign errors when finding factor pairs of c. When factoring x² + bx + c, if c is positive and b is negative, both factors are negative: x² - 7x + 12 = (x - 3)(x - 4), NOT (x + 3)(x + 4). If c is negative, the factors have opposite signs.
• Forgetting to multiply a × c when factoring ax² + bx + c. For 2x² + 7x + 3, first multiply 2 × 3 = 6, then find numbers that multiply to 6 and add to 7. Skipping the a × c step leads to incorrect factorisations.
• Stopping factorisation too early with the difference of squares. After getting (x² + 4)(x² - 4) from x⁴ - 16, remember that x² - 4 can be factored further to (x + 2)(x - 2). Always check if any factor is itself a difference of squares.

Next Steps

Next up: 01-08-quadratic-equations.md