Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 1 - Quadratics and polynomials

Quadratics and Polynomials | Year 10 Mathematics

Connect quadratic expressions, factors, roots and graphs so algebra becomes useful for modelling.

Updated 2026-06-15 - 4 min read

A quadratic is an expression or relationship where the highest power of the variable is 2. Year 10 students need to connect three views of a quadratic:

  • the expanded expression
  • the factorised expression
  • the graph

Expanded and factorised forms

The expression $x^2 + 5x + 6$ is in expanded form.

It can be factorised as $(x + 2)(x + 3)$ because:

$(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$

The factorised form is useful because it tells us when the expression equals zero.

If $(x + 2)(x + 3) = 0$, then either:

  • $x + 2 = 0$, so $x = -2$
  • $x + 3 = 0$, so $x = -3$

So the graph crosses the x-axis at $x = -2$ and $x = -3$.

Graph features

A quadratic graph is called a parabola. Useful features include:

  • x-intercepts: where the graph crosses the x-axis
  • y-intercept: where the graph crosses the y-axis
  • turning point: the minimum or maximum point
  • axis of symmetry: the vertical line through the turning point

For $y = x^2 + 5x + 6$, the y-intercept is 6 because when $x = 0$, $y = 6$.

The x-intercepts are $-2$ and $-3$ because the factorised form is $(x + 2)(x + 3)$.

Choosing the best form

Different forms answer different questions.

Expanded form is useful for seeing the y-intercept quickly. In $y = x^2 + 5x + 6$, the constant term is 6, so the graph crosses the y-axis at 6.

Factorised form is useful for seeing roots. In $y = (x + 2)(x + 3)$, the roots are $x = -2$ and $x = -3$.

Vertex or turning-point form, such as $y = (x - 1)^2 - 4$, is useful for seeing the turning point. This note does not require every vertex-form method, but it is helpful to know why senior maths often rewrites quadratics in different forms.

The Year 10 habit is strategic form choice. Do not expand everything automatically. Ask what information the question wants.

Why this matters for senior readiness

In senior Mathematics, quadratics appear in functions, modelling, rates, optimisation and graph interpretation. The useful Year 10 habit is not memorising one method. It is switching forms when a different form gives better information.

Quadratics also train a communication habit that matters in senior assessment: connect algebraic steps to features of the situation. If a root is negative in a time question, explain why it is not meaningful. If a turning point is a maximum height, name the height and the time. If a factorisation gives two possible values, decide which values suit the context.

This is why showing working is not only about marks. It records the decisions that make the answer mathematically and contextually sensible.

Quick check

  1. Factorise $x^2 + 7x + 12$.
  2. What are the roots of $x^2 + 7x + 12 = 0$?
  3. What is the y-intercept of $y = x^2 + 7x + 12$?

Answers:

  1. $(x + 3)(x + 4)$
  2. $x = -3$ and $x = -4$
  3. 12

Transfer task

Find a situation that could create a quadratic relationship, such as area, projectile height or profit. State what the variable represents, what the intercepts or turning point could mean, and one limitation of the model.

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