Sec 2 Quadratic Equations and Graphs | Factorisation, Roots and Parabolas

Secondary 2 Quadratic Equations and Graphs is a lower-secondary bridge into one of the central algebraic structures of upper-secondary Mathematics: relationships involving x². School sequencing can vary, so this guide teaches the mathematics without assuming every Sec 2 course introduces every quadratic method at the same time.

Students searching for Sec 2 quadratic equations, quadratic graphs, factorising quadratics, roots or parabola often treat equations and graphs as separate topics. They are the same relationship viewed differently: the roots of the equation are the x-values where the graph crosses the x-axis.

This page is the canonical Quadratics bridge owner under the Secondary Mathematics Topic Library. It depends on Expansion and Factorisation and Linear Graphs.

Quick answer: what makes an expression quadratic?

A quadratic contains a highest power of x².

General form: ax²+bx+c, where a ≠ 0.

Examples: x²+5x+6, 2x²−3x+1, x²−9.

Quadratic equation

A quadratic equation sets a quadratic expression equal to zero or another value.

Example: x²+5x+6=0.

Solving by factorisation

x²+5x+6 = (x+2)(x+3).

So (x+2)(x+3)=0.

A product is zero when at least one factor is zero.

Therefore x=−2 or x=−3.

Zero-product principle

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

This is what makes factorisation useful for solving quadratic equations.

Worked example 1

Solve: x²−7x+12=0.

Factorise: (x−3)(x−4)=0.

Solutions: x=3 or x=4.

Difference of two squares

x²−a² = (x−a)(x+a).

Example: x²−25 = (x−5)(x+5).

So x²−25=0 gives x=5 or x=−5.

Quadratic graphs

A graph of y=ax²+bx+c is a parabola.

a > 0: opens upward.
a < 0: opens downward.

The graph is curved because the rate of change itself changes.

Roots and x-intercepts

Where the graph crosses the x-axis, y=0.

Therefore the x-intercepts solve ax²+bx+c=0.

This is the direct connection between the graph and equation.

Worked example 2: graph meaning

For y=x²−5x+6 = (x−2)(x−3), the roots are x=2 and x=3.

So the graph crosses the x-axis at (2,0) and (3,0).

The y-intercept

At x=0, y=c.

For y=x²−5x+6, y-intercept = 6, so the graph passes through (0,6).

Symmetry of a parabola

A parabola has a vertical axis of symmetry through its turning point.

If the roots are 2 and 6, the axis of symmetry lies halfway between them at x=4.

This midpoint idea is a useful graph check.

Table of values

A table helps sketch a quadratic graph accurately. Choose x-values around the likely turning region, calculate y, plot the points and draw a smooth curve.

Do not connect quadratic points with straight segments as if the relationship were linear.

Worked table

For y=x²−4:

x=−3 → y=5
x=−2 → y=0
x=0 → y=−4
x=2 → y=0
x=3 → y=5

The graph is symmetric about x=0.

Linear versus quadratic

Linear: constant gradient, straight graph.
Quadratic: changing gradient, curved parabola.

Students should learn to recognise the structural difference, not only the visual one.

Common errors

factorises incorrectly
forgets the ± pair from two factors
sets only one factor equal to zero
confuses y-intercept with roots
joins quadratic points with straight lines
assumes every quadratic has two distinct real roots
forgets school/course sequencing can vary and rushes into advanced methods too early

The quadratics audit

Can the expression be factorised?
Is the equation arranged with zero on one side?
Have both factors been set to zero?
Do the roots match the graph’s x-intercepts?
Does the sign of a match the opening direction?

Practice

Practice 1

Question: Factorise x²+6x+8

Answer: (x+2)(x+4)

Practice 2

Question: Solve x²−9=0

Answer: x=±3

Practice 3

Question: Solve x²−5x+6=0

Answer: x=2 or 3

Practice 4

Question: For y=x²+3x−4, y-intercept

Answer: −4

Practice 5

Question: Roots 1 and 5: axis of symmetry

Answer: x=3

Frequently asked questions

Are quadratics always taught in exactly the same Secondary year?

School and course sequencing can vary. This page is a lower-secondary bridge owner in Atlas, not a claim of one universal teaching calendar.

Why factorise a quadratic?

Factorisation reveals factors that can be set to zero to find roots.

What do roots mean on the graph?

They are the x-values where y=0, so the graph crosses or touches the x-axis.

How is a quadratic graph different from a line?

Its gradient changes, so the graph curves.

Where does this sit in Atlas?

This is the canonical Sec 2 Quadratic Equations and Graphs bridge owner under Secondary Mathematics Topic Library.

The final Quadratics rule

Treat factorisation, equations and graphs as one system. The factor form reveals roots; the roots locate x-intercepts; the graph makes the quadratic relationship visible.

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