Homeschool Guide: These lesson plans are a guide for parents. Content may contain errors — always cross-reference with official exam board specifications.

solving quadratic equations

FoundationHigherAll Boards

4 detailed 50-minute lessons with teaching scripts, worked examples, parent guides, and assessment criteria.

Fastmail

Lesson Overview

Total Lessons: 4
Tier: Foundation and Higher
Duration: 50 minutes per lesson (200 minutes total)
Exam Boards: AQA, Edexcel, OCR, Eduqas, CCEA

Learning Objectives

Prerequisites

Materials & Equipment

Lesson 1: Introduction: solving quadratic equations

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Write down everything you already know about solving quadratic equations. Then check against the key terms: Quadratic equation, Methods. Use a mini-whiteboard or paper.

Main Content (35 minutes)

Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: Quadratic equation, Methods.
If stuck: Re-read the revision notes (link above), then break the content into smaller steps.
Extension: See the Stretch & Challenge ideas in Lesson 4.
Teaching Script (35 mins):
Mins 0-5 - Hook: "Today: solving quadratic equations. By the end you will be able to answer exam questions on it unaided. It connects to the rest of Mathematics because the ideas here recur across the spec."
Mins 5-20 - Direct Instruction: Work through the core ideas below one at a time; after each, ask your student to explain it back in their own words.
Mins 20-30 - Guided Practice: Model the worked example together, then let your student attempt the first practice question with guidance.
Mins 30-35 - Independent Practice: 2-3 practice questions from Lesson 3 below, with immediate feedback.
First Look

Start with the revision notes summary, then attempt: Solve: x² + 7x + 12 = 0

Plenary (5 minutes)

Check Out

Your student states one thing they learned and one question they still have about solving quadratic equations.

Lesson 2: Core Concepts: solving quadratic equations

Duration: 50 minutes

Starter Activity (5 minutes)

Review Previous Lesson

Quick recap: write 3 key points from Lesson 1 on solving quadratic equations. Check them against the notes below.

Main Content (35 minutes)

Quadratic equation: An equation of the form ax² + bx + c = 0 where a ≠ 0.
Methods: Factorising (all levels) Quadratic formula (Higher) Completing the square (Higher) Graphically (reading x-intercepts)

Practice (10 minutes)

Q: Solve: x² + 7x + 12 = 0

Answer: x = -3 or x = -4

Plenary (5 minutes)

Explain Back

Your student teaches the key points back to you without looking. Fill any gaps immediately.

Lesson 3: Application: solving quadratic equations

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Recall the key terms: Quadratic equation, Methods. Define each in one sentence.

Main Content (35 minutes)

Parent/Teacher Guide: Let your student attempt each question alone first, then compare with the model answer. Award method marks for correct working even if the final answer is wrong.

Q1: Solve: x² + 7x + 12 = 0

Answer: x = -3 or x = -4

Q2: Solve: x² - 16 = 0

Answer: x = 4 or x = -4

Q3: Solve: x² - 3x - 10 = 0

Answer: x = 5 or x = -2

Q4: Solve using the formula: x² + 5x + 2 = 0 (to 2 d.p.)

Answer: x = -0.44 or x = -4.56

Q5: Solve by completing the square: x² + 10x + 6 = 0

Answer: x = -5 + √19 or x = -5 - √19

Q6: How many solutions does x² + 6x + 9 = 0 have?

Answer: One solution (x = -3, repeated root)

Plenary (5 minutes)

Error Review

Review any questions answered incorrectly. Identify whether the error was knowledge, method, or reading the question.

Lesson 4: Exam Practice: solving quadratic equations

Duration: 50 minutes

Starter Activity (5 minutes)

Command Words

Review what these command words require: state (one point), describe (say what happens), explain (say why), compare (both sides), evaluate (judgement).

Main Content (35 minutes)

Extended Answer

Extended question: Extended Answer 6 marks: A rectangle has sides (x + 2) cm and (x - 1) cm. Its area is 20 cm². (a) Form and solve a quadratic equation. (b) Find the dimensions. (c) Verify that your answer is correct. <div class="

(a) (x + 2)(x - 1) = 20 → x² + x - 2 = 20 → x² + x - 22 = 0 x = -1 ± √(1 + 88) ⁄ 2 = -1 ± √89 ⁄ 2 . x must be positive, so x = -1 + 9.43 ⁄ 2 ≈ 4.22 (b) Length ≈ 6.22 cm, Width ≈ 3.22 cm (c) 6.22 × 3.22 ≈ 20.03 ≈ 20 cm² ✓ Mark scheme: (a) 2 marks for equation and solving. (b) 2 marks for dimensions. (c) 2 marks for verification.

Exam Tips: Always rearrange to ax² + bx + c = 0 first | Try factorising first - it's often the quickest method | Use the quadratic formula when factorising is difficult | Check answers by substituting back | Remember: b² - 4ac tells you how many solutions
Common Errors: Watch Out! 1. Wrong: x² = 9 so x = 3 Correct: x = 3 OR x = -3 (don't forget the negative solution) 2. Wrong: x² + 5x + 6 = 2 factorises as (x + 2)(x + 3) = 2 Correct: Must rearrange to = 0 first: x² + 5x + 4 = 0, then (x + 1)(x + 4) = 0 3. Wrong: In the quadratic formula, using c = 6 for x² + 5x = -6 Correct: Rearrange first: x² + 5x + 6 = 0, so c = 6 (positive, not -6)
AO3 - Reasoning & Interpretation: Reasoning and Interpretation A company's profit P in £1000s is modelled by P = -x² + 10x - 16, where x is the price in pounds. (a) Find the prices that give zero profit. (b) What price maximises profit, and what is the maximum? (c) Explain why not all prices between the break-even points give positive profit. Answers: (a) -x² + 10x - 16 = 0 → (x-2)(x-8) = 0 → x = £2 or x = £8. (b) Turning point at x = 5: P = -25 + 50 - 16 = 9 (£9000). (c) This is incorrect — actually all prices between £2 and £8 DO give positive profit since the parabola opens downward. The profit is negative only outside this range.
Stretch & Challenge (Grade 8-9):
  • Synoptic links: explain how solving quadratic equations connects to another Mathematics topic you have studied
  • Real-world: research one real-world use or example of solving quadratic equations
  • Critical: "What are the limitations of the models used in solving quadratic equations?"

Plenary (5 minutes)

Assessment Criteria
  • Got it: Confident explanation + correct worked examples
  • Getting there: Main points OK, needs support with detail
  • Not yet: Confused on key concepts - re-run Lesson 2

Homework & Consolidation

Recommended Resources

🎓 Smart Lesson (Guided)