Solving Quadratic Equations by Factorising

Active learning builds fluency in algebraic manipulation, which is essential when solving systems involving linear and quadratic equations. Students need repeated, scaffolded practice to transfer skills from basic factorising to solving for coordinates, making hands-on and collaborative tasks ideal for this topic.

• Year 11
• Mathematics
• 3 activities
• 15 min–30 min

Learning Objectives

1. Factorise quadratic expressions of the form ax^2 + bx + c and x^2 + bx + c to find the roots.
2. Explain the relationship between the factors of a quadratic expression and the solutions (roots) of the corresponding equation.
3. Calculate the roots of quadratic equations by applying factorisation methods.
4. Compare the efficiency of solving quadratic equations by factorising versus completing the square for different types of equations.
5. Determine whether a quadratic equation has two distinct real roots, one repeated real root, or no real roots based on its factorised form.

Think-Pair-Share

⏱ 15 min·Pairs

Think-Pair-Share: The Substitution Strategy

Provide students with a linear and quadratic pair. Individually, they identify which variable is easiest to isolate; in pairs, they perform the substitution and discuss why isolating 'y' might be simpler than 'x' in specific cases.

Prepare & details

Analyze how factorising a quadratic expression reveals its roots.

Facilitation Tip: During Think-Pair-Share, assign specific roles so students verbalise each step of the substitution process to their partner before solving together.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Inquiry Circle

⏱ 30 min·Small Groups

Inquiry Circle: Intersection Hunt

Give small groups sets of equations and a large coordinate grid. They must algebraically solve the systems and then plot them to verify if their solutions match the physical intersections on the graph.

Prepare & details

Compare the efficiency of factorising versus other methods for specific quadratic forms.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Peer Teaching

⏱ 20 min·Pairs

Peer Teaching: Error Detectives

Present a pre-written 'failed' solution containing common expansion or sign errors. Students work in pairs to find the mistake, explain the correct step to their partner, and present the fix to the class.

Prepare & details

Explain why a quadratic equation can have two, one, or zero real solutions.

Setup: Presentation area at front, or multiple teaching stations

Materials: Topic assignment cards, Lesson planning template, Peer feedback form, Visual aid supplies

• Understand
• Apply
• Analyze
• Create
• Self-Management
• Relationship Skills

Teaching This Topic

Teachers should model the substitution process slowly, writing each step visibly and speaking the reasoning aloud. Research shows that students benefit from seeing errors corrected in real time, so deliberately introduce a common mistake and work through its resolution as a class. Avoid rushing to the final answer; emphasise the importance of checking that found solutions satisfy both original equations.

What to Expect

Successful students will confidently substitute a linear equation into a quadratic, factorise correctly, solve for both variables, and state intersection points with coordinates. They will also recognise when factorising is appropriate and when other methods are more efficient.

Watch Out for These Misconceptions

Common Misconception

During Think-Pair-Share, watch for students who solve the quadratic equation but stop before calculating the corresponding y-coordinates.

What to Teach Instead

Provide a printed checklist with the steps ‘Substitute, Expand, Factorise, Solve for x, Find y, Write coordinates’ and have partners check off each step as they work.

Common Misconception

During Collaborative Investigation, watch for students who incorrectly expand binomials during substitution, such as treating (x + 3)^2 as x^2 + 9.

What to Teach Instead

Hand out algebra tiles or grid method templates so students can model the expansion physically before recording the algebraic form, reinforcing the need for the middle term.

Assessment Ideas

Exit Ticket

After Think-Pair-Share, collect each pair’s solutions with full coordinates for x^2 + 5x + 6 = 0 and 2x^2 - 7x + 3 = 0 to check both factorising and coordinate completion.

Quick Check

During Peer Teaching, circulate and ask each student to explain one step of the error correction process; listen for accurate linkage between factorised form and roots.

Discussion Prompt

After Collaborative Investigation, pose the discussion prompt and ask students to justify their answers using examples from their investigation graphs or equations.

Extensions & Scaffolding

• Challenge: Provide a system where the quadratic is not in standard form (e.g., y = x^2 + 3x - 10 and y = 2x + 1) and ask students to rearrange it before substitution.
• Scaffolding: Offer partially completed substitution templates with gaps for students to fill in the linear expression and expanded quadratic terms.
• Deeper exploration: Explore how changing the constant term in the quadratic affects the number and coordinates of intersection points using graphing software.

Key Vocabulary

Quadratic Expression	An algebraic expression of the form ax^2 + bx + c, where a, b, and c are constants and a is not equal to zero.
Factorise	To express a quadratic expression as a product of two or more simpler expressions (factors).
Root (or Solution)	A value of the variable (usually x) that makes a quadratic equation equal to zero.
Zero Product Property	If the product of two or more factors is zero, then at least one of the factors must be zero. This is essential for solving factorised equations.