📐 Concept diagram

01-08 - Quadratic Equations

Phase: 1 | Subject: 01-08 Prerequisites: 01-07-quadratic-expressions.md (factorising quadratics) Next subject: 01-09-polynomials.md

Learning Objectives

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

1. Solve quadratic equations by factorising
2. Use the quadratic formula for any quadratic equation
3. Calculate and interpret the discriminant
4. Complete the square to solve quadratics
5. Convert between standard form, vertex form, and factored form

Core Content

What is a Quadratic Equation?

A quadratic equation has the form:

$ax² + bx + c = 0$ where a ≠ 0

The highest power of x is 2. Solutions are called roots or x-intercepts — where the parabola crosses the x-axis.

Zero Product Property

If AB = 0, then A = 0 OR B = 0.

This is the key to solving by factorising.

Example: (x - 3)(x + 2) = 0 - Either x - 3 = 0 → x = 3 - Or x + 2 = 0 → x = -2

Solutions: x = 3 or x = -2

Solving by Factorising

Steps: 1. Ensure equation is in form ax² + bx + c = 0 2. Factorise the left side 3. Set each factor equal to 0 and solve

Example: Solve x² - 5x + 6 = 0

1. Already in standard form
2. Factorise: (x - 2)(x - 3) = 0
3. x - 2 = 0 → x = 2
4. x - 3 = 0 → x = 3

Solutions: x = 2 or x = 3

Example: Solve 2x² + 7x + 3 = 0

1. Factorise: (2x + 1)(x + 3) = 0
2. 2x + 1 = 0 → x = -1/2
3. x + 3 = 0 → x = -3

Solutions: x = -1/2 or x = -3

The Quadratic Formula

⚠️ THIS IS CRITICAL — the quadratic formula is one of the most-used formulas in all of mathematics. You will use it constantly in later phases: optimisation problems, physics, engineering, and it underpins the discriminant analysis used in linear algebra and ML.

For ANY quadratic equation ax² + bx + c = 0:

$x = (-b ± √(b² - 4ac)) / (2a)
$

The ± means there are TWO solutions (usually).

Common pitfall — forgetting the ±: The ± symbol is NOT optional. It gives both solutions. Omitting it gives only ONE root. Always compute both: one with + and one with -.

Common pitfall — equation must = 0: The quadratic formula requires the equation to be in the form ax² + bx + c = 0. If your equation is not equal to zero (e.g., x² + 3x = 10), rearrange it first: x² + 3x - 10 = 0.

Example: Solve x² - 4x - 5 = 0

1. a = 1, b = -4, c = -5
2. x = (4 ± √((-4)² - 4(1)(-5))) / 2
3. x = (4 ± √(16 + 20)) / 2
4. x = (4 ± √36) / 2
5. x = (4 ± 6) / 2
6. x = (4 + 6)/2 = 5 or x = (4 - 6)/2 = -1

Solutions: x = 5 or x = -1

The Discriminant

The expression under the square root: Δ = b² - 4ac

Example: Discriminant of x² + 2x + 2 = 0

Δ = 2² - 4(1)(2) = 4 - 8 = -4 < 0

No real solutions. The parabola never crosses the x-axis.

Completing the Square

Convert ax² + bx + c to a(x - h)² + k form.

Steps: 1. If a ≠ 1, factor out a from first two terms 2. Take half of b, square it, add and subtract inside 3. Rewrite as perfect square

Example: Complete the square for x² + 6x + 2

1. Take half of 6: 3. Square it: 9.
2. Add and subtract 9: x² + 6x + 9 - 9 + 2
3. Rewrite: (x + 3)² - 7

To solve x² + 6x + 2 = 0: 1. (x + 3)² - 7 = 0 2. (x + 3)² = 7 3. x + 3 = ±√7 4. x = -3 ± √7

Example: Complete the square for 2x² + 8x + 3

1. Factor out 2: 2(x² + 4x) + 3
2. Half of 4: 2. Square: 4.
3. 2(x² + 4x + 4 - 4) + 3
4. 2((x + 2)² - 4) + 3
5. 2(x + 2)² - 8 + 3
6. 2(x + 2)² - 5

Vertex Form

From completing the square: a(x - h)² + k

• (h, k) is the vertex (turning point) of the parabola
• h = -b/(2a)
• k = f(h) = value at the vertex

Example: y = x² - 4x + 3

1. Complete square: (x - 2)² - 1
2. Vertex: (2, -1)
3. Opens upward (a = 1 > 0), so vertex is minimum

Key Terms

• 01 08 Quadratic Equations
• Completing the Square
• Correct: A)
• Correct: C)
• Discriminant
• Example 1: Solve by factorising x² - 3x - 10 = 0
• Example 2: Solve using quadratic formula 3x² + 2x - 1 = 0
• Example 3: Complete the square for x² + 4x + 1 = 0
• Solving by Factorising
• The Discriminant
• The Quadratic Formula
• Vertex Form

Worked Examples

Example 1: Solve by factorising x² - 3x - 10 = 0

1. Find numbers that multiply to -10 and add to -3: -5 and 2
2. (x - 5)(x + 2) = 0
3. x - 5 = 0 → x = 5
4. x + 2 = 0 → x = -2

Solutions: x = 5, x = -2

Example 2: Solve using quadratic formula 3x² + 2x - 1 = 0

1. a = 3, b = 2, c = -1
2. x = (-2 ± √(4 - 4(3)(-1))) / 6
3. x = (-2 ± √(4 + 12)) / 6
4. x = (-2 ± √16) / 6
5. x = (-2 ± 4) / 6
6. x = 2/6 = 1/3 or x = -6/6 = -1

Solutions: x = 1/3, x = -1

Example 3: Complete the square for x² + 4x + 1 = 0

1. (x² + 4x + 4) - 4 + 1 = 0
2. (x + 2)² - 3 = 0
3. (x + 2)² = 3
4. x + 2 = ±√3
5. x = -2 ± √3

Solutions: x ≈ -2 + 1.732 = -0.268 or x ≈ -2 - 1.732 = -3.732

Quiz

Q1: What does the concept of Completing the Square primarily refer to in this subject?

A) The definition and application of Completing the Square B) A visual representation of Completing the Square C) A computational error related to Completing the Square D) A historical anecdote about Completing the Square

Correct: A)

• If you chose A: Completing the Square is defined as: the definition and application of completing the square. The other options describe different aspects that are not the primary focus. Correct!
• If you chose B: This is incorrect. Completing the Square is defined as: the definition and application of completing the square. The other options describe different aspects that are not the primary focus.
• If you chose C: This is incorrect. Completing the Square is defined as: the definition and application of completing the square. The other options describe different aspects that are not the primary focus.
• If you chose D: This is incorrect. Completing the Square is defined as: the definition and application of completing the square. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of Discriminant?

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

Correct: A)

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

Q3: Which statement about Solving by Factorising is TRUE?

A) Solving by Factorising is an advanced topic beyond this subject's scope B) Solving by Factorising is not related to this subject C) Solving by Factorising is mentioned only as a historical footnote D) Solving by Factorising is a fundamental concept covered in this subject

Correct: D)

• If you chose A: This is incorrect. Solving by Factorising is a fundamental concept covered in this subject. This subject covers Solving by Factorising as part of its core content.
• If you chose B: This is incorrect. Solving by Factorising is a fundamental concept covered in this subject. This subject covers Solving by Factorising as part of its core content.
• If you chose C: This is incorrect. Solving by Factorising is a fundamental concept covered in this subject. This subject covers Solving by Factorising as part of its core content.
• If you chose D: Solving by Factorising is a fundamental concept covered in this subject. This subject covers Solving by Factorising 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) 0 C) The inverse of the correct answer D) An unrelated numerical value

Correct: B)

• If you chose A: This is incorrect. The worked examples show that the result is 0. The other options represent common errors.
• If you chose B: The worked examples show that the result is 0. The other options represent common errors. Correct!
• If you chose C: This is incorrect. The worked examples show that the result is 0. The other options represent common errors.
• If you chose D: This is incorrect. The worked examples show that the result is 0. The other options represent common errors.

Q5: How are Solving by Factorising and The Discriminant related?

A) Solving by Factorising is a special case of The Discriminant B) Solving by Factorising and The Discriminant are completely unrelated topics C) Solving by Factorising and The Discriminant are closely related concepts D) Solving by Factorising is the inverse of The Discriminant

Correct: C)

• If you chose A: This is incorrect. Both Solving by Factorising and The Discriminant are covered in this subject as interconnected topics.
• If you chose B: This is incorrect. Both Solving by Factorising and The Discriminant are covered in this subject as interconnected topics.
• If you chose C: Both Solving by Factorising and The Discriminant are covered in this subject as interconnected topics. Correct!
• If you chose D: This is incorrect. Both Solving by Factorising and The Discriminant are covered in this subject as interconnected topics.

Q6: What is a common pitfall when working with The Quadratic Formula?

A) The Quadratic Formula is always computed the same way in all contexts B) A common mistake is confusing The Quadratic Formula with a similar concept C) The Quadratic Formula has no common misconceptions D) The main error with The Quadratic Formula is using it when it is not needed

Correct: B)

• If you chose A: This is incorrect. Students often confuse The Quadratic Formula with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: Students often confuse The Quadratic Formula with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose C: This is incorrect. Students often confuse The Quadratic Formula with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: This is incorrect. Students often confuse The Quadratic Formula with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply Vertex Form?

A) Use Vertex Form only in pure mathematics contexts B) Apply Vertex Form to solve problems in this subject's domain C) Vertex Form is not practically useful D) Avoid Vertex Form unless explicitly instructed

Correct: B)

• If you chose A: This is incorrect. Vertex Form is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: Vertex Form is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose C: This is incorrect. Vertex Form is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: This is incorrect. Vertex Form is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. Solve: x² - 7x + 12 = 0
   Click for answer

(x - 3)(x - 4) = 0. x = 3 or x = 4.

1. Solve: x² + 5x + 6 = 0
   Click for answer

(x + 2)(x + 3) = 0. x = -2 or x = -3.

1. Solve: 2x² - 5x - 3 = 0
   Click for answer

Factorise: (2x + 1)(x - 3) = 0. x = -1/2 or x = 3.

1. Solve using formula: x² - 2x - 8 = 0
   Click for answer

a=1, b=-2, c=-8. x = (2 ± √(4 + 32))/2 = (2 ± 6)/2. x = 4 or x = -2.

1. Find discriminant of 2x² + 3x + 1 = 0 and interpret
   Click for answer

Δ = 9 - 8 = 1 > 0. Two distinct real roots.

1. Complete the square: x² + 8x + 5 = 0
   Click for answer

(x + 4)² - 16 + 5 = 0, (x + 4)² = 11, x = -4 ± √11.

1. Solve: 3x² + 6x + 2 = 0
   Click for answer

a=3, b=6, c=2. x = (-6 ± √(36 - 24))/6 = (-6 ± √12)/6 = (-6 ± 2√3)/6 = -1 ± √3/3.

Summary

Key takeaways:

• Quadratic equation: ax² + bx + c = 0, a ≠ 0
• Factorising: use zero product property — set each factor to 0
• Quadratic formula works for ALL quadratics: x = (-b ± √(b²-4ac)) / 2a
• Discriminant Δ = b² - 4ac tells you about the nature of roots
• Completing the square reveals the vertex form a(x-h)² + k
• Vertex (h, k) is the turning point: h = -b/(2a)

Pitfalls

• Forgetting the ± in the quadratic formula. The ± symbol gives BOTH solutions. Computing only x = (-b + √Δ)/(2a) and forgetting to also compute (-b - √Δ)/(2a) loses one root. Always write both and compute separately.
• Not setting the equation to zero before using the quadratic formula. The formula requires ax² + bx + c = 0. If you have x² + 3x = 10, rearrange to x² + 3x - 10 = 0 first. Plugging a, b, c from the non-zero form gives completely wrong results.
• Misinterpreting the discriminant. Δ > 0 means two distinct real roots, Δ = 0 means one repeated root, Δ < 0 means no real roots (two complex). A common error is thinking Δ = 0 means no solution — it actually means one solution where the parabola touches the x-axis.
• Losing a negative sign when substituting negative b into the quadratic formula. For x² - 4x - 5 = 0, b = -4, so -b = +4. The formula is (-b ± √Δ)/(2a). Plugging -4 directly instead of -(-4) = 4 is a frequent sign error.
• In completing the square, forgetting to balance the equation when factoring out a coefficient other than 1. For 2x² + 8x + 3, after writing 2(x² + 4x + 4 - 4) + 3, you must multiply the -4 by 2 when moving it outside: 2(x + 2)² - 8 + 3 = 2(x + 2)² - 5. The -4 inside the parentheses gets multiplied by the factor of 2.

Next Steps

Next up: 01-09-polynomials.md