## Quadratics: $ax^2 + bx + c = 0$

A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a \neq 0$. The graph of a quadratic is a parabola.

## Solving Quadratic Equations

There are several methods to solve quadratic equations:

• Factorising: If the quadratic can be factorised into two brackets, set each bracket to zero to find the solutions (roots). E.g., $x^2 + 5x + 6 = 0 \Rightarrow (x+2)(x+3) = 0 \Rightarrow x = -2, x = -3$. This method is quick when applicable.
• Quadratic Formula: Always works for any quadratic equation. Memorise: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Substitute the values of $a, b, c$ from your equation. The term $b^2 - 4ac$ is the discriminant.
• If $b^2 - 4ac > 0$, there are two distinct real roots.
• If $b^2 - 4ac = 0$, there is one repeated real root.
• If $b^2 - 4ac < 0$, there are no real roots.
• Completing the Square: This method transforms $ax^2 + bx + c$ into the form $a(x+p)^2 + q$. It's useful for solving equations, finding the turning point (vertex) of a parabola $(-p, q)$, and sketching graphs. To solve, rearrange to $(x+p)^2 = k$, then take the square root.

## Graphs of Quadratics

A quadratic graph is a parabola.

• If $a > 0$, the parabola is U-shaped (minimum point).
• If $a < 0$, the parabola is n-shaped (maximum point).
• The roots (solutions) are where the graph crosses the x-axis ($y=0$).
• The y-intercept is where $x=0$, which is $(0, c)$.
• The axis of symmetry passes through the turning point. Its equation is $x = -p$ from the completed square form $a(x+p)^2+q$, or $x = \frac{-b}{2a}$.

## Simultaneous Equations

Solving simultaneous equations means finding values for variables that satisfy all equations at once.

• Linear-Linear: Two equations, both linear.
• Elimination: Multiply equations to make coefficients of one variable the same, then add or subtract to eliminate that variable.
• Substitution: Rearrange one equation to make a variable the subject, then substitute into the other equation.
• Linear-Quadratic: One linear, one quadratic equation.
• Substitution is the primary method. Rearrange the linear equation to make one variable the subject, then substitute this into the quadratic equation. This will result in a single quadratic equation to solve for one variable, then substitute back to find the other. You will typically find two pairs of solutions, representing the two intersection points of the line and the parabola.

## Surds

Surds are irrational numbers that involve a root symbol (e.g., √2, √3). They cannot be expressed as a simple fraction.

• Simplifying Surds: Look for square factors. `√(ab) = √a × √b`.
• *Example:* `√50 = √(25 × 2) = √25 × √2 = 5√2`.
• Adding and Subtracting Surds: You can only add or subtract surds if the number inside the root is the same.
• *Example:* `3√5 + 7√5 = 10√5`. `√12 + √3 = 2√3 + √3 = 3√3`.
• Multiplying Surds: `√a × √b = √ab`. `(a√x)(b√y) = ab√(xy)`.
• *Example:* `√3 × √7 = √21`. `(2√3)(5√2) = 10√6`.
• Rationalising the Denominator: This means removing any surds from the denominator.
• If the denominator is `√a`, multiply the numerator and denominator by `√a`.
• *Example:* `1/√5 = (1 × √5)/(√5 × √5) = √5/5`.
• If the denominator is `a + √b` or `a - √b`, multiply by its conjugate (`a - √b` or `a + √b`). Remember `(x+y)(x-y) = x² - y²`.
• *Example:* `1/(2 + √3) = (1 × (2 - √3))/((2 + √3)(2 - √3)) = (2 - √3)/(2² - (√3)²) = (2 - √3)/(4 - 3) = 2 - √3`.

## Indices (Powers)

Indices are a shorthand for repeated multiplication. `a^n` means `a` multiplied by itself `n` times.

• Rules of Indices:
• Multiplication: `a^m × a^n = a^(m+n)`
• Division: `a^m / a^n = a^(m-n)`
• Power of a Power: `(a^m)^n = a^(mn)`
• Zero Power: `a^0 = 1` (for `a ≠ 0`)
• Negative Power: `a^(-n) = 1/a^n`
• Fractional Power: `a^(1/n) = n√a` (the nth root of a)
• General Fractional Power: `a^(m/n) = (n√a)^m` or `n√(a^m)`

• Solving Index Equations: Make the bases the same.
• *Example:* `2^x = 16` becomes `2^x = 2^4`, so `x = 4`.

## Standard Form

Standard form (or scientific notation) is a way to write very large or very small numbers concisely.

• Format: `A × 10^n`, where `1 ≤ A < 10` and `n` is an integer.
• `A` is a number between 1 and 10 (not including 10).
• `n` is an integer (positive for large numbers, negative for small numbers).
• *Example (large number):* `3,400,000 = 3.4 × 10^6` (decimal point moved 6 places left).
• *Example (small number):* `0.0000078 = 7.8 × 10^-6` (decimal point moved 6 places right).
• Calculations:
• Multiplication: Multiply the `A` parts, add the `n` parts. `(A × 10^n) × (B × 10^m) = (A × B) × 10^(n+m)`.
• Division: Divide the `A` parts, subtract the `n` parts. `(A × 10^n) / (B × 10^m) = (A / B) × 10^(n-m)`.
• Always ensure the final answer is in correct standard form (adjust `A` and `n` if necessary).

## Ratio

Ratios compare parts of a whole or different quantities. They can be simplified by dividing all parts by their highest common factor, just like fractions. For example, 6:9 simplifies to 2:3. To divide a quantity in a given ratio, add the parts of the ratio to find the total number of 'shares'. Then divide the total quantity by this sum to find the value of one share, and multiply by each part of the ratio. For example, to divide £60 in the ratio 2:3, total shares = 2+3=5. One share = £60/5 = £12. So, £24 and £36.

## Proportion

Direct Proportion means that as one quantity increases, the other increases at the same rate. If y is directly proportional to x, then y = kx, where k is the constant of proportionality. If y is directly proportional to x², then y = kx². The graph of direct proportion (y=kx) is a straight line through the origin.

Inverse Proportion means that as one quantity increases, the other decreases. If y is inversely proportional to x, then y = k/x. If y is inversely proportional to x², then y = k/x². To solve proportion problems, first find k using a given pair of values, then use k to find unknown values.

## Compound Measures

Compound measures combine two or more units. Key examples include:

• Speed: Distance ÷ Time (e.g., m/s, km/h)
• Density: Mass ÷ Volume (e.g., g/cm³, kg/m³)
• Pressure: Force ÷ Area (e.g., N/m², Pascals)

Remember to ensure units are consistent before calculations. For example, if speed is in km/h, distance must be in km and time in hours.

## Percentage Change and Compound Interest

To calculate a percentage increase or decrease, use a multiplier. For a 10% increase, multiply by 1.10. For a 10% decrease, multiply by 0.90.

Compound Interest involves calculating interest on the original amount plus any accumulated interest from previous periods. The formula is: Original Amount × (1 + percentage increase/100)^number of periods.

Reverse Percentages are used when you know the final amount after a percentage change and need to find the original amount. Divide the final amount by the multiplier. For example, if a price of £120 includes 20% VAT, the original price is £120 / 1.20 = £100.

Growth and Decay problems often use the compound interest formula structure.

## Pythagoras' Theorem

• Applies only to right-angled triangles.
• The longest side is called the hypotenuse (opposite the right angle).
• Formula: a² + b² = c², where 'c' is the hypotenuse and 'a', 'b' are the other two sides.
• To find the hypotenuse: c = √(a² + b²).
• To find a shorter side: a = √(c² - b²).
• Used in 2D and 3D problems (e.g., finding the diagonal of a cuboid or the height of a cone).

## Right-Angled Trigonometry (SOH CAH TOA)

• Used for finding unknown sides or angles in right-angled triangles.
• Label sides relative to the angle (θ) you are working with: Opposite (O), Adjacent (A), Hypotenuse (H).
• SOH: Sin(θ) = Opposite / Hypotenuse
• CAH: Cos(θ) = Adjacent / Hypotenuse
• TOA: Tan(θ) = Opposite / Adjacent
• To find a side, use the appropriate ratio and rearrange the formula.
• To find an angle, use the inverse trigonometric functions: sin⁻¹(O/H), cos⁻¹(A/H), tan⁻¹(O/A).
• Remember exact trigonometric values for 0°, 30°, 45°, 60°, 90°.

## Non-Right-Angled Trigonometry

• For triangles without a right angle, use the Sine Rule or Cosine Rule.

Sine Rule

• Use when you have a matching pair of a side and its opposite angle, plus one other side or angle.
• To find a side: a / sin A = b / sin B = c / sin C
• To find an angle: sin A / a = sin B / b = sin C / c

Cosine Rule

• Use when you have all three sides (to find an angle) or two sides and the included angle (to find the third side).
• To find a side: a² = b² + c² - 2bc cos A (where A is the angle opposite side a).
• To find an angle: cos A = (b² + c² - a²) / 2bc

## Area of Any Triangle

• Formula: Area = ½ab sin C, where 'a' and 'b' are two sides and 'C' is the included angle between them.

## Trigonometric Graphs

• y = sin x: S-shaped wave, period 360°, passes through (0,0), max at 90°, min at 270°.
• y = cos x: C-shaped wave, period 360°, passes through (0,1), max at 0°/360°, min at 180°.
• y = tan x: Repeating curve, period 180°, asymptotes at 90°, 270°, etc. Values repeat every 180°.

## Circle Theorems

Circle theorems describe relationships between angles, tangents, chords, and radii in a circle.

• The angle at the centre is twice the angle at the circumference subtended by the same arc.
• Angles in the same segment are equal.
• The angle in a semicircle is always 90°.
• Opposite angles of a cyclic quadrilateral (a quadrilateral with all vertices on the circumference) sum to 180°.
• A tangent to a circle is perpendicular to the radius at the point of contact.
• The alternate segment theorem states that the angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment.
• Tangents drawn from an external point to a circle are equal in length.
• The perpendicular from the centre to a chord bisects the chord.

## Vectors

Vectors represent movement with both magnitude (size) and direction. They can be written as column vectors, e.g., $\begin{pmatrix} x \\ y \end{pmatrix}$, or using i and j components, e.g., $x\mathbf{i} + y\mathbf{j}$.

• Vector addition/subtraction: Add/subtract corresponding components. Geometrically, vectors are added head-to-tail.
• Scalar multiplication: Multiply each component by the scalar. This changes the magnitude but not the direction (unless the scalar is negative, which reverses direction).
• Magnitude of a vector $\begin{pmatrix} x \\ y \end{pmatrix}$ is $\sqrt{x^2 + y^2}$ (Pythagoras' theorem).
• Parallel vectors are scalar multiples of each other.
• Position vectors start from the origin. If $\vec{OA} = \mathbf{a}$ and $\vec{OB} = \mathbf{b}$, then $\vec{AB} = \mathbf{b} - \mathbf{a}$.
• Points A, B, C are collinear if $\vec{AB}$ is a scalar multiple of $\vec{BC}$ (or $\vec{AC}$) and they share a common point.

## Transformations

Transformations move or change the size of shapes.

• Translation: Moves a shape without rotating, reflecting, or resizing it. Defined by a vector $\begin{pmatrix} x \\ y \end{pmatrix}$.
• Rotation: Turns a shape around a fixed point. Defined by a centre of rotation, an angle, and a direction (clockwise/anticlockwise).
• Reflection: Flips a shape over a line. Defined by a line of reflection (e.g., $y=x$, $x=0$).
• Enlargement: Changes the size of a shape. Defined by a centre of enlargement and a scale factor. A scale factor greater than 1 makes the shape bigger; between 0 and 1 makes it smaller. A negative scale factor enlarges and rotates the shape by 180 degrees through the centre.
• You must be able to describe a transformation fully or perform one given its parameters.

## Probability Basics

• Probability measures the likelihood of an event, ranging from 0 (impossible) to 1 (certain).
• P(Event) = (Number of favourable outcomes) / (Total number of possible outcomes).
• The sum of probabilities of all possible outcomes for an experiment is 1.
• Mutually Exclusive Events: Events that cannot happen at the same time. If A and B are mutually exclusive, P(A or B) = P(A) + P(B).
• Independent Events: The outcome of one event does not affect the outcome of another. If A and B are independent, P(A and B) = P(A) × P(B).
• Tree Diagrams are useful for visualising sequences of events, especially when probabilities change (dependent events).
• Venn Diagrams illustrate relationships between sets of events, helping to calculate probabilities like P(A and B), P(A or B), and P(A').
• Relative Frequency (Experimental Probability) = (Number of times an event occurs) / (Total number of trials). It approximates theoretical probability as the number of trials increases.
• Expected Number of Occurrences = P(event) × Number of trials.

## Statistics: Data Handling and Representation

• Types of Data: Qualitative (non-numerical, e.g., colour), Quantitative (numerical). Quantitative data can be Discrete (countable, e.g., number of siblings) or Continuous (measurable, e.g., height).
• Histograms: Used for continuous data with unequal class widths. The y-axis represents frequency density (Frequency / Class Width), not frequency. The area of each bar is proportional to the frequency.
• Cumulative Frequency Diagrams: Plot cumulative frequency against the upper class boundary. Used to estimate the median, lower quartile (Q1), upper quartile (Q3), and interquartile range (IQR = Q3 - Q1).
• Box Plots: A visual summary of the five-number summary: minimum, Q1, median, Q3, maximum.
• Scatter Graphs: Show the relationship (correlation) between two variables. Correlation can be positive, negative, or non-existent. A line of best fit can be drawn for estimation (interpolation or less reliable extrapolation). Remember, correlation does not imply causation.

## Statistics: Measures and Sampling

• Measures of Central Tendency: These indicate the 'average' or 'centre' of the data.
• Mean: Sum of all values / Number of values. For grouped data, estimate using midpoints: Σ(midpoint × frequency) / Σfrequency.
• Median: The middle value when data is ordered. For grouped data, it's estimated from a cumulative frequency diagram.
• Mode: The most frequent value or modal class.
• Measures of Spread: These describe how spread out the data is.
• Range: Maximum value - Minimum value.
• Interquartile Range (IQR): Q3 - Q1. It's less affected by outliers than the range.
• Sampling: The process of selecting a subset from a larger population.
• A sample should be representative of the population to avoid bias.
• Random Sampling: Every member of the population has an equal chance of selection.
• Stratified Sampling: Divides the population into subgroups (strata) and takes a proportional random sample from each stratum to ensure fair representation.