Outcomes
A student:

 MA5.31WM
uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures

 MA5.32WM
generalises mathematical ideas and techniques to analyse and solve problems efficiently

 MA5.33WM
uses deductive reasoning in presenting arguments and formal proofs

 MA5.37NA
solves complex linear, quadratic, simple cubic and simultaneous equations, and rearranges literal equations
Content
 Students:
 Solve complex linear equations involving algebraic fractions

solve a range of linear equations, including equations that involve two or more fractions,
eg \( \frac{2x5}{3}  \frac{x+7}{5} = 2 \), \(\quad \frac{y1}{4}  \frac{2y+3}{3} = \frac{1}{2} \)
 Solve a wide range of quadratic equations derived from a variety of contexts (ACMNA269)
 solve equations of the form \( ax^2 + bx + c = 0 \) by factorisation and by 'completing the square'
 use the quadratic formula \( x = \frac{b \pm \sqrt{b^2  4ac}}{2a} \) to solve quadratic equations
 solve a variety of quadratic equations, eg \( 3x^2 = 4 \), \( x^2  8x 4 = 0 \), \( x(x4) = 4 \), \( (y2)^2 = 9 \)
 choose the most appropriate method to solve a particular quadratic equation (Problem Solving)
 check the solutions of quadratic equations by substituting
 identify whether a given quadratic equation has real solutions, and if there are real solutions, whether they are or are not equal
 predict the number of distinct real solutions for a particular quadratic equation (Communicating, Reasoning)
 connect the value of \( b^2  4ac \) to the number of distinct solutions of \( ax^2 + bx + c = 0 \) and explain the significance of this connection (Communicating, Reasoning)
 solve quadratic equations resulting from substitution into formulas
 create quadratic equations to solve a variety of problems and check solutions
 explain why one of the solutions to a quadratic equation generated from a word problem may not be a possible solution to the problem (Communicating, Reasoning)
 substitute a pronumeral to simplify higherorder equations so that they can be seen to belong to general categories and then solve the equations, eg substitute \(u\) for \(x^2\) to solve \( x^4  13x^2 + 36 = 0 \) for \(x\)
 Solve simple cubic equations
 determine that for any value of \(k\) there is a unique value of \(x\) that solves a simple cubic equation of the form \(ax^3 = k\) where \(a \ne 0\)
 explain why cubic equations of the form \(ax^3 = k\) where \(a \ne 0\) have a unique solution (Communicating, Reasoning)
 solve simple cubic equations of the form \(ax^3 = k\), leaving answers in exact form and as decimal approximations
 Rearrange literal equations

change the subject of formulas, including examples from other strands and other learning areas,
eg make a the subject of \(v = u + at\), make r the subject of \( \frac{1}{x} = \frac{1}{r} + \frac{1}{s} \), make b the subject of \( x = \sqrt{b^2  4ac} \)
 determine restrictions on the values of variables implicit in the original formula and after rearrangement of the formula, eg consider what restrictions there would be on the variables in the equation \( Z = ax^2 \) and what additional restrictions are assumed if the equation is rearranged to \( x = \sqrt{\frac{Z}{a}} \) (Communicating, Reasoning)
 Solve simultaneous equations, where one equation is nonlinear, using algebraic and graphical techniques, including the use of digital technologies

use analytical methods to solve a variety of simultaneous equations, where one equation is nonlinear,
eg \( \begin{cases} y = x^2 \\ y = x \end{cases} \), \( \begin{cases} y = x^2  x  2\\ y = x + 6 \end{cases} \), \( \begin{cases} y = x + 5\\ y = \frac{6}{x} \end{cases} \)
 choose an appropriate method to solve a pair of simultaneous equations (Problem Solving, Reasoning)
 solve pairs of simultaneous equations, where one equation is nonlinear, by finding the point of intersection of their graphs using digital technologies
 determine and explain that some pairs of simultaneous equations, where one equation is nonlinear, may have no real solutions (Communicating, Reasoning)
Background Information
The derivation of the quadratic formula can be demonstrated for more capable students.
Language
In Stage 6, the term 'discriminant' is introduced for the expression \(b^2  4ac\). It is not expected that students in Stage 5 will use this term; however, teachers may choose to introduce the term at this stage if appropriate.