Key Exam Notes: Algebra, Equations, Indices, Measurement, and Quadratics

Algebra Review

• Like Terms: Add/subtract like terms only (e.g., $2x + 4x + y = 6x + y$).

• Multiplication: $3x \times 2y = 6xy$

• Division: $4ab \div (2b) = 2a$

• Expansion: $a(b + c) = ab + ac$, $a(b - c) = ab - ac$

• Solving Linear Equations: Use inverse operations and collect like terms.

Algebraic Fractions

• Addition/Subtraction: Find a common denominator.

  • E.g., $\frac{2}{x + 3} + \frac{1}{3} = \frac{3(x + 3) + 2(x + 1)}{6} = \frac{5x + 11}{6}$

• Solving Equations: Combine fractions, then multiply both sides by the denominator.

Linear Inequalities

• Solving: Solve as linear equations, but reverse the inequality symbol when multiplying or dividing by a negative number.

• Graphing: Represent solutions on a number line.

Graphs of Linear Relations

• Form: $y = mx + c$, where $m$ is the gradient and $c$ is the y-intercept.

• Graphing: Use the gradient and y-intercept, or locate x-intercept $(y = 0)$ and y-intercept $(x = 0)$.

Regions on the Cartesian Plane

• Inequalities represent regions above or below a line.

• Shade above the line for $\geqslant$, use a dashed line for > .

• Test $(0, 0)$ to decide which region to shade for inequalities like 2x + 3y < 6.

Midpoint and Length

• Midpoint: $M = (\frac{x<em>1 + x</em>2}{2}, \frac{y<em>1 + y</em>2}{2})$

• Length/Distance: $d = \sqrt{(x<em>2 - x</em>1)^2 + (y<em>2 - y</em>1)^2}$

Parallel and Perpendicular Lines

• Parallel: Same gradient (e.g., $y = 2x + 7$ and $y = 2x - 4$).

• Perpendicular: Gradients $m<em>1$ and $m</em>2$ satisfy $m<em>1 \times m</em>2 = -1$

Finding a Line Equation

• Use $y = mx + c$ or $y - y<em>1 = m(x - x</em>1)$.

• Find the gradient and y-intercept, or substitute a point if $c$ is unknown.

Simultaneous Equations

• Solve using substitution (when one pronumeral is the subject) or elimination (adding/subtracting multiples to eliminate a variable).

Exponential Equations

• If $a^x = a^y$, then $x = y$.

Scientific Notation

• Numbers in the form * a x 10^m*, where 1 \leqslant a < 10 or -10 < a \leqslant -1, and $m$ is an integer.

Index Laws

• Multiply: $a^m \times a^n = a^{m+n}$

• Divide: $a^m \div a^n = a^{m-n}$

• Power of a Power: $(a^m)^n = a^{mn}$

• Product to a Power: $(ab)^m = a^m b^m$

• Quotient to a Power: $(a/b)^m = a^m / b^m$

Zero and Negative Indices

• Zero index: $a^0 = 1, a \ne 0$

• Negative index: $a^{-m} = \frac{1}{a^m}$

Fractional Indices

• $a^{\frac{1}{n}} = \sqrt[n]{a}$

Logarithms

• Definition: $log_a y = x$ is equivalent to $a^x = y$, where a > 0

• Log law for addition: $log<em>a x + log</em>a y = log_a (xy)$

• Log law for subtraction: $log<em>a x - log</em>a y = log_a (\frac{x}{y})$

• Log law involving powers: $log<em>a x^n = n log</em>a x$

Compound Interest

• Formula: $A = P(1 + \frac{r}{100})^n$

  • where:

  • A (amount): total value

  • P (principal): initial amount

  • r (interest rate): per period

  • n (period): number of periods

Chapter 4: Measurement and Surds

Simplifying Surds (10A)

• Use the highest square factor to simplify surds (e.g., $\sqrt{12} = 2\sqrt{3}$).

• Rationalise the denominator to express with a whole number in the denominator (e.g., $\frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}$).

• Add/Subtract like surds only.

• Multiply/Divide surds using the rule $\sqrt{x} \times=\sqrt{xy}$

Pythagoras theorem

$c^2= a^2 + b^2$

Circle

Circumference =$2\pi r = \pi d$
Area= $\pi r^2$

Chapter 5: Quadratic expressions and equations

Quadratic formula

If $ax^2 + bx + c = 0$ , then
$x=\frac{-b \pm \sqrt{b^2 -4ac}}{2a}$

The discriminant

$\Delta = b^2 - 4ac$

• If \Delta>0 Two solutions
  -If $\Delta=0$ One solutions
  -If \Delta<0 Zero solutions
  ## Factorising
  Factorising and Difference of two squares
  $a^2 - b^2 = (a-b)(a+b)$

Applications

1 Define the variable

2. Set up the equation

3. Solve by factorising and using the Null Factor Law or quadratic formula

4. Determine the suitable answer(s)