NSW Syllabuses

# Mathematics K–10 - Stage 5.3 - Number and Algebra Equations §

## Outcomes

#### A student:

• MA5.3-1WM

uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures

• MA5.3-2WM

generalises mathematical ideas and techniques to analyse and solve problems efficiently

• MA5.3-3WM

uses deductive reasoning in presenting arguments and formal proofs

• MA5.3-7NA

solves complex linear, quadratic, simple cubic and simultaneous equations, and rearranges literal equations

## Content

• solve a range of linear equations, including equations that involve two or more fractions,
eg $$\frac{2x-5}{3} - \frac{x+7}{5} = 2$$,  $$\quad \frac{y-1}{4} - \frac{2y+3}{3} = \frac{1}{2}$$
• 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 $$3x^2 = 4$$,   $$x^2 - 8x -4 = 0$$,   $$x(x-4) = 4$$,   $$(y-2)^2 = 9$$
• 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 higher-order 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 non-linear, 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 non-linear,
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 non-linear, 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 non-linear, 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.