Quadratics — NSW Year 10 Advanced Maths
Expanding and factorising quadratics, then solving them by factorising, the formula and completing the square. Ten subtopics and 200 practice questions with worked solutions, aligned to NESA outcomes MA5-ALG-P-02 and MA5-EQU-P-02.
Quadratics in NSW Year 10 Advanced
A quadratic expression has \(x^2\) as its highest power, such as \(6x^2+x-2\). Year 10 Advanced students expand and factorise quadratics, including non-monic trinomials and special products, then solve quadratic equations by factorising, by completing the square and with the quadratic formula \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\), and use them to model real problems.
The ten subtopics below start with expanding, then factorising monic and non-monic trinomials, special binomial products and factorising by completing the square, then solving by factorising, by the quadratic formula and by completing the square, and finish with higher order equations and problem solving. Quadratics is one of the 16 topics in NSW Year 10 Advanced Maths and sits within the Number and Algebra strand (NESA outcomes MA5‑ALG‑P‑01, MA5‑ALG‑P‑02 and MA5‑EQU‑P‑02). It leads to parabolas, polynomials and Year 11 Advanced.
Subtopics
Quadratics: common questions
How do you factorise a non-monic quadratic?
Multiply the first and last coefficients, find two numbers with that product and the middle coefficient as their sum, split the middle term and factorise in pairs. For \(6x^2+x-2\): \(6\times(-2)=-12\), use 4 and \(-3\), so \(6x^2+4x-3x-2=(3x+2)(2x-1)\).
What is the quadratic formula?
For \(ax^2+bx+c=0\), \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\). It works for every quadratic equation, including ones that do not factorise, such as \(x^2+9x+19=0\).
How do you know how many solutions a quadratic has?
Look at \(b^2-4ac\). If it is positive there are two real solutions, if it is zero there is one, and if it is negative there are none. For \(4x^2+12x+9=0\), \(b^2-4ac=144-144=0\), so \(x=-1.5\) is the only solution.
What is completing the square?
Add the square of half the coefficient of \(x\) to make a perfect square. For \(x^2-2x-1=0\): \(x^2-2x+1=2\), so \((x-1)^2=2\) and \(x=1\pm\sqrt{2}\).
Which NESA outcomes cover Quadratics?
Factorising monic trinomials is MA5‑ALG‑P‑01, the special products and non-monic trinomials are MA5‑ALG‑P‑02, and solving quadratic equations by any method, including word problems and equations that reduce to quadratics, is MA5‑EQU‑P‑02, all Path content in Stage 5 of the NSW Mathematics K–10 Syllabus (2022).