Sec 2 G3 Maths Bukit Timah | Quadratic Equations

Secondary 2 G3 Mathematics Tutor Bukit Timah | Quadratic Equations by Factorisation — Did you know? A quadratic equation can have two correct answers. That is not a mistake: it is what often happens when we turn one algebraic expression into two factors.

Quick answer: how do you solve a quadratic by factorisation?

First rearrange the equation so it equals zero. Factorise the quadratic into a product such as (x + 2)(x + 3) = 0. The zero-product rule means at least one factor must be zero, so x = −2 or x = −3. Check both values in the original equation.

Here is what I look for in a student’s working: not just correct brackets, but an explanation of why each bracket can be equated to zero. Understanding that one principle makes unfamiliar quadratic questions less intimidating.

Why can one equation have two roots?

Suppose two numbers are multiplied and the result is zero. Either the first number is zero, or the second is zero, or both are zero. We use this basic fact when a quadratic equation is written as a product equal to zero.

Quadratic equation	Factorised equation	Roots
x² + 5x + 6 = 0	(x + 2)(x + 3) = 0	−2 and −3
x² − 25 = 0	(x − 5)(x + 5) = 0	5 and −5
2x² − 7x + 3 = 0	(2x − 1)(x − 3) = 0	1/2 and 3
x² − 6x + 9 = 0	(x − 3)² = 0	3, repeated root

An important detail: the zero-product rule is valid when the product is zero. If (x + 2)(x + 3) = 12, we cannot set each bracket equal to zero. First rearrange the equation correctly.

Worked example 1: simple quadratic factorisation

Question: Solve x² + 5x + 6 = 0.

Look for two integers with product +6 and sum +5. They are +2 and +3.
Factorise: (x + 2)(x + 3) = 0.
Use the zero-product rule: x + 2 = 0 or x + 3 = 0.
Therefore x = −2 or x = −3.
Check x = −2: 4 − 10 + 6 = 0. Check x = −3: 9 − 15 + 6 = 0.

Did you know? The signs of the terms inside the brackets and the signs of the roots are opposite in this example. A student who understands the equation x + 2 = 0 will not need to guess the sign of the answer.

Worked example 2: the coefficient of x² is not one

Question: Solve 2x² − 7x + 3 = 0.

Try (2x − 1)(x − 3).
Expand to check: 2x² − 6x − x + 3 = 2x² − 7x + 3.
Set 2x − 1 = 0, giving x = 1/2.
Set x − 3 = 0, giving x = 3.
Both satisfy the original equation, so both are valid roots.

Checking by expansion is a reliable defence against incorrect middle-term signs. This reverse operation connects the solving skill with expansion and factorisation rather than treating them as separate topics.

Worked example 3: difference of two squares

Question: Solve x² − 49 = 0.

Recognise 49 as 7².
Factorise x² − 7² = (x − 7)(x + 7).
Set x − 7 = 0 or x + 7 = 0.
Answer: x = 7 or x = −7.

It is easy to write only +7 when mentally taking a square root. The two factors make the negative root visible.

Worked example 4: a rectangle word problem

A rectangle has width x cm and length (x + 3) cm. Its area is 40 cm². Find its dimensions.

Write x(x + 3) = 40 using the area formula.
Expand and rearrange: x² + 3x − 40 = 0.
Factorise: (x + 8)(x − 5) = 0.
Algebra gives x = −8 or x = 5.
A width cannot be negative, so take x = 5 cm and length = 8 cm.
Check the context: 5 × 8 = 40 cm².

The equation has two mathematical solutions, but the word problem asks for a real width. Rejecting a negative physical length is not a failure to report a root; it is part of interpreting the question.

How is a quadratic root connected to its graph?

The graph y = x² + 5x + 6 crosses the horizontal x-axis wherever y = 0. Those crossing x-values are precisely the roots −3 and −2. The algebra and the graph are telling the same story.

But not all quadratic equations have two distinct real roots. For example, x² − 6x + 9 = (x − 3)² gives a repeated root x = 3. Its graph just touches the x-axis at that point. See the companion guide on quadratic graphs and turning points.

Four common mistakes and the correct fix

Forgetting ‘= 0’: rearrange before using the zero-product rule.
Incorrect signs in factors: re-expand them to test the original coefficients.
Stopping at the factorised brackets: solve both linear factors explicitly.
Keeping an impossible answer to a word problem: check what x represents before reporting dimensions, quantities or costs.

Self-check questions and answers

A. Solve x² − x − 12 = 0. Factorise (x − 4)(x + 3) = 0. Answers: 4, −3.

B. Solve x² − 16 = 0. Answers: 4, −4.

C. Solve x² + 8x + 16 = 0. Answer: −4, a repeated root.

D. Solve 3x² − 12x = 0. Factorise 3x(x − 4) = 0. Answers: 0, 4.

A focused fifteen-minute revision plan

Three minutes: re-expand two sets of proposed factors to check signs.
Four minutes: solve two equations using the zero-product rule.
Four minutes: solve a difference-of-squares equation and a repeated-root equation.
Four minutes: write the quadratic for a rectangle, solve it and explain any rejected value.

A useful tutor check is to remove the heading that says ‘quadratic equations’. Can the student recognise which expression must first be made zero? That is the point where chapter practice becomes independent Mathematics.

How might a Bukit Timah Secondary 2 G3 Maths tutor help?

In a three-pupil tutorial, one learner may need help recognising a factor pair, another with signed numbers, and another with translating a word problem into an equation. The tutor should diagnose those differences and ask every pupil to attempt the changed question themselves.

The unchanged eduKateSG Secondary 1 Mathematics small-group teaching benchmark describes three-student classes, weekly 1.5-hour tutorials, close feedback and carefully guided corrections. Its published teaching address is 8 Fourth Avenue, Singapore 268674, near Sixth Avenue MRT. Bukit Timah here refers to families researching nearby tuition, not a claim of additional branch premises. Confirm Secondary 2 G3 timetable and availability with the centre.

G3 Mathematics and school assessment priorities

G3 is a Mathematics subject level under Full Subject-Based Banding. A Secondary 2 pupil should focus on the school’s actual chapter order and weighted assessment topics. For curriculum context see the MOE G2/G3 Mathematics syllabus and a Singapore school’s Secondary 2 Mathematics chapter outline.

The Singapore–Cambridge SEC first takes place in 2027, but the full upper-secondary syllabus is not the immediate Secondary 2 assessment list. Parents should check the teacher’s instructions for what their child is doing now.

Parent FAQ: What if my child can factorise but cannot solve?

That often means they have learned the bracket pattern but not the zero-product principle. Ask the child to explain why one factor must be zero and then solve a new example. Avoid introducing more complex methods before this idea is reliable.

Parent FAQ: Is a negative solution always wrong?

No. The roots −2 and −3 of a pure algebraic equation are perfectly valid. A negative solution becomes inappropriate only when the problem represents something that cannot be negative, such as a physical side length.

Read the related Mathematics guides

Original teaching reference and consultation

Programme benchmark: eduKateSG small-group Mathematics tutorials. Ask about Secondary 2 G3 class placement through eduKate Singapore or WhatsApp.

Educational editorial review: 24 October 2026. All worked Mathematics examples are original educational illustrations, not copied examination questions. Current lesson and centre details must be verified.