Mathematics · Year 10 · Algebraic Structure and Manipulation · Autumn Term

Solving Quadratic Equations by Factorising

Solving quadratic equations by factorising and applying the null factor law.

• National Curriculum Attainment Targets
• GCSE: Mathematics - Algebra

About This Topic

Solving quadratic equations by factorising requires students to rewrite expressions such as x² + 7x + 12 = 0 into (x + 3)(x + 4) = 0, then apply the null factor law: if the product of two factors equals zero, each factor must equal zero separately. Year 10 students identify number pairs that multiply to the constant term and add to the x coefficient, starting with integer roots before addressing problems like x² - 5x + 6 = 0. This process links algebraic rearrangement to practical applications, such as finding points where parabolas cross the x-axis in projectile motion.

Within the GCSE Mathematics Algebra unit, factorising builds core manipulation skills and prepares students for completing the square or the quadratic formula. They evaluate its efficiency for quadratics with rational roots and predict solutions directly from factorised forms, fostering strategic thinking about equation types.

Active learning excels with this topic through tactile matching games and group verification. Students handle physical cards or digital drags to pair equations with roots, debate efficiencies, and correct peers' work. These approaches make abstract rules concrete, boost procedural fluency, and reduce errors from isolation.

Key Questions

1. Explain why the null factor law is fundamental to solving quadratic equations by factorising.
2. Evaluate the efficiency of factorising compared to other methods for specific quadratic equations.
3. Predict the roots of a quadratic equation given its factorised form.

Learning Objectives

• Factorise quadratic expressions of the form ax² + bx + c into the product of two linear factors.
• Apply the null factor law to solve quadratic equations that have been factorised.
• Calculate the roots of quadratic equations by first factorising them and then applying the null factor law.
• Compare the efficiency of solving quadratic equations by factorising versus using the quadratic formula for equations with integer or simple rational roots.

Before You Start

Expanding Double Brackets

Why: Students need to be proficient in multiplying two linear expressions to understand how to reverse this process during factorisation.

Identifying Factors of Integers

Why: The process of factorising quadratics relies on finding pairs of numbers that multiply to a specific value (the constant term).

Basic Algebraic Manipulation

Why: Students should be comfortable with rearranging simple algebraic expressions and solving linear equations.

Key Vocabulary

Quadratic Equation	An equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a is not equal to zero.
Factorising	The process of expressing a polynomial as a product of its factors, typically simpler polynomials.
Null Factor Law	A rule stating that if the product of two or more factors is zero, then at least one of the factors must be zero.
Roots	The values of the variable (usually x) that make a quadratic equation true; also known as solutions or zeros.

Watch Out for These Misconceptions

Common Misconception

All quadratics factorise easily over integers.

What to Teach Instead

Many have irrational roots, requiring quadratic formula instead. Sorting activities where groups classify factorisable versus non-factorisable equations help students test discriminants intuitively and choose methods wisely.

Common Misconception

After factorising, solve the factors as new quadratics.

What to Teach Instead

Set each linear factor to zero directly via null factor law. Peer teaching in relay tasks clarifies this, as students verbalise steps and catch when teammates overcomplicate.

Common Misconception

Ignore signs when finding factor pairs.

What to Teach Instead

Pairs must match both product and sum accurately. Error hunts in pairs reveal sign flips quickly, with collaborative rewriting reinforcing the rule through immediate feedback.

Active Learning Ideas

Card Sort: Equation to Roots

Prepare cards with unsolved quadratics, factorised forms, graphs, and root pairs. Small groups sort and match sets on tables, then create their own cards to swap with another group. End with a class share-out of tricky matches.

35 min·Small Groups

Error Hunt: Faulty Solutions

Provide worksheets with five factorised solutions containing common errors like sign flips or ignored null factor steps. Pairs identify mistakes, explain corrections, and rewrite correctly. Groups present one to the class.

25 min·Pairs

Relay Factorise: Chain Challenge

Divide class into teams lined up. First student factorises a quadratic on the board, tags next for null factor law application, and so on until roots found. Fastest accurate team wins; repeat with varied equations.

40 min·Small Groups

Prediction Pairs: Roots First

Give factorised forms like (x - 2)(x + 5) = 0; pairs predict roots, expand to verify, then solve reverse from expanded form. Switch roles and compare efficiencies.

30 min·Pairs

Real-World Connections

• Engineers designing suspension bridges use quadratic equations to model the parabolic shape of the main cables, and solving these equations helps determine stress points and material requirements.
• In physics, when analyzing projectile motion, quadratic equations describe the path of an object under gravity. Finding where the equation equals zero (i.e., the roots) tells us when the object hits the ground.

Assessment Ideas

Quick Check

Present students with three quadratic equations: one easily factorised (e.g., x² + 5x + 6 = 0), one requiring a common factor (e.g., 2x² + 6x = 0), and one that does not factorise easily over integers (e.g., x² + 2x - 5 = 0). Ask students to solve the first two by factorising and state why the third is more efficiently solved by another method.

Exit Ticket

Give students the factorised equation (x - 3)(x + 5) = 0. Ask them to: 1. State the roots of the equation. 2. Write the expanded quadratic equation. 3. Explain in one sentence how the null factor law was used.

Discussion Prompt

Pose the question: 'When solving a quadratic equation, is factorising always the best first step?' Facilitate a class discussion where students share examples of when factorising is efficient and when it might be time-consuming or impossible, leading them to consider other methods like completing the square or the quadratic formula.

Frequently Asked Questions

What is the null factor law in quadratic equations?

The null factor law states that if a product equals zero, at least one factor is zero. For (x + 2)(x - 3) = 0, solutions are x = -2 or x = 3. Teach it via expansions and contractions: students multiply roots back to quadratics, solidifying why it works for all factorised forms in GCSE Algebra.

How do you factorise x² + 5x + 6 efficiently?

Find pairs multiplying to 6 and adding to 5: 2 and 3. So x² + 5x + 6 = (x + 2)(x + 3). Practice with timers in pairs builds speed; compare to formula for b² - 4ac = 25 - 24 = 1, confirming roots -2 and -3 match.

When is factorising more efficient than the quadratic formula?

Use factorising for integer or simple rational roots, where pairs are obvious, saving time over formula calculations. For x² - 7x + 12 = 0, (x - 3)(x - 4) is instant versus formula steps. Class debates on examples develop judgement for exam strategy.

How can active learning help students master quadratic factorising?

Activities like card sorts and relays engage kinesthetic learners, making factor pairs and null factor law memorable through movement and talk. Groups debate matches, reducing solo errors by 30-40% in trials, while prediction tasks link roots to graphs visually. This builds confidence for GCSE problems over passive worksheets.