## Algebra, Quadratics & Simultaneous Equations

This topic covers essential algebraic techniques for manipulating expressions, solving various types of equations, and understanding functions, all crucial for the AQA GCSE Higher Maths exam.

Quadratics

A quadratic equation is an equation of the form ax² + bx + c = 0, where a ≠ 0. Its graph is a parabola (a U-shaped curve).

You can solve quadratic equations using several methods:

• Factorising: If possible, express the quadratic as a product of two linear factors. Set each factor to zero to find the solutions (roots). E.g., x² + 5x + 6 = 0 becomes (x+2)(x+3) = 0, so x = -2 or x = -3.
• Quadratic Formula: This formula works for *any* quadratic equation ax² + bx + c = 0. The solutions are given by x = [-b ± √(b² - 4ac)] / 2a.
• Completing the Square: Transform x² + bx + c into the form (x+p)² + q. This method is useful for solving, and it directly reveals the turning point (vertex) of the parabola, which is at (-p, q).

Simultaneous Equations

These involve finding values for multiple variables that satisfy two or more equations at the same time. The solutions represent the intersection points of the graphs of the equations.

• Linear Simultaneous Equations: Both equations are linear (no powers higher than 1). Solve using:
• Substitution: Rearrange one equation to make a variable the subject (e.g., y = ...), then substitute this expression into the other equation.
• Elimination: Multiply one or both equations so that the coefficients of one variable are the same (or additive inverses), then add or subtract the equations to eliminate that variable.
• Linear and Quadratic Simultaneous Equations: Always use substitution. Rearrange the linear equation to make a variable the subject, then substitute this expression into the quadratic equation. This will result in a quadratic equation which you then solve. The solutions will be the coordinates of the intersection points of the line and the curve.

Algebraic Fractions

These involve algebraic expressions in fractions. You need to be able to:

• Simplify: Factorise both the numerator and denominator, then cancel any common factors.
• Add/Subtract: Find a common denominator before adding or subtracting the numerators.
• Multiply/Divide: For multiplication, multiply the numerators and multiply the denominators. For division, invert the second fraction and then multiply.

Functions

A function is a rule that maps each input value to exactly one output value.

• Function notation: f(x) represents the output of the function f when the input is x.
• Inverse functions (f⁻¹(x)): An inverse function "undoes" the original function. To find f⁻¹(x), let y = f(x), swap x and y, then rearrange to make y the subject again.
• Composite functions (fg(x) or f(g(x))): This means applying one function after another. For fg(x), first apply the function g to x, then apply the function f to the result of g(x).

Algebraic Proof

This involves using logical algebraic steps to demonstrate that a mathematical statement is always true for all possible values of the variables involved. For example, proving that the sum of two consecutive odd numbers is always a multiple of 4.

## Surds

Surds are irrational numbers that involve a root symbol (e.g., √2, √3). They cannot be expressed as a simple fraction. You often need to simplify and manipulate them.

• Simplifying Surds: Look for square factors. For example, √12 = √(4 × 3) = √4 × √3 = 2√3.
• Adding and Subtracting Surds: You can only add or subtract 'like' surds (surds with the same number under the root). First, simplify them if necessary. E.g., 3√2 + 5√2 = 8√2. But √2 + √3 cannot be simplified.
• Multiplying Surds: Multiply the numbers outside the root and the numbers inside the root separately. E.g., (2√3) × (4√5) = (2×4)√(3×5) = 8√15.
• Rationalising the Denominator: This means removing the surd from the denominator.
• If the denominator is a single surd (e.g., 1/√a), multiply the numerator and denominator by √a. E.g., 1/√3 = (1×√3)/(√3×√3) = √3/3.
• If the denominator is a binomial surd (e.g., 1/(a + √b)), multiply the numerator and denominator by its conjugate (a - √b). E.g., 1/(2 + √3) = (1×(2 - √3))/((2 + √3)×(2 - √3)) = (2 - √3)/(4 - 3) = 2 - √3.

## Indices

Indices (or powers) show how many times a number is multiplied by itself. The laws of indices are crucial for manipulating expressions.

• Multiplication: a^m × a^n = a^(m+n)
• Division: a^m ÷ a^n = a^(m-n)
• Powers of Powers: (a^m)^n = a^(mn)
• Zero Index: a^0 = 1 (any non-zero number to the power of 0 is 1)
• Negative Index: a^(-n) = 1/a^n (a negative power means the reciprocal)
• Fractional Indices:
• a^(1/n) = n√a (the nth root of a)
• a^(m/n) = (n√a)^m or n√(a^m)

## Standard Form

Standard form (or scientific notation) is a way to write very large or very small numbers concisely. A number in standard form is written as A × 10^n, where:

• 1 ≤ A < 10 (A is a number between 1 and 10, including 1 but not 10)
• n is an integer (positive for large numbers, negative for small numbers).

• Converting to Standard Form: Move the decimal point until A is between 1 and 10, then count the number of places moved to find n.
• E.g., 34500 = 3.45 × 10^4 (moved 4 places left, so n=4)
• E.g., 0.000067 = 6.7 × 10^-5 (moved 5 places right, so n=-5)
• Calculations: When multiplying or dividing numbers in standard form, multiply/divide the A parts and add/subtract the powers of 10. For addition/subtraction, convert to ordinary numbers or adjust powers of 10 to be the same.

## Ratio

• A ratio compares parts of a whole or different quantities. It shows how much of one thing there is compared to another. For example, a ratio of 1:3 means for every one part of the first item, there are three parts of the second.
• Simplifying ratios: Divide all parts of the ratio by their highest common factor (e.g., 20:30 simplifies to 2:3 by dividing by 10). Ensure all quantities are in the same units before simplifying.
• Sharing in a given ratio: To share an amount, say £100, in the ratio 2:3:

1. Add the parts of the ratio: 2 + 3 = 5 parts.

2. Divide the total amount by the sum of parts: £100 / 5 = £20 per part.

3. Multiply each ratio part by this value: 2 x £20 = £40 and 3 x £20 = £60.

• Ratio problems: Can involve finding missing quantities or comparing ratios, often requiring algebraic methods for more complex scenarios.

## Proportion

• Direct Proportion: Two quantities, A and B, are directly proportional if A = kB, where k is the constant of proportionality. As one quantity increases, the other increases at a constant rate. The graph of direct proportion is a straight line passing through the origin.
• Inverse Proportion: Two quantities, A and B, are inversely proportional if A = k/B, where k is the constant of proportionality. As one quantity increases, the other decreases. The graph of inverse proportion is a reciprocal curve.
• Solving proportion problems:

1. Write the proportionality relationship (e.g., y ∝ x, y ∝ 1/x, y ∝ x²).

2. Change the proportionality to an equation using 'k' (e.g., y = kx, y = k/x, y = kx²).

3. Use the given pair of values to calculate the constant 'k'.

4. Use the complete equation with the calculated 'k' to find any unknown values.

## Compound Measures

• A compound measure combines two or more basic units of measurement.
• Speed: The rate at which an object covers distance. Formula: Speed = Distance / Time. Units include m/s, km/h.
• Density: The mass per unit volume of a substance. Formula: Density = Mass / Volume. Units include g/cm³, kg/m³.
• Pressure: The force applied perpendicular to the surface of an object per unit area. Formula: Pressure = Force / Area. Units include N/m² (Pascals, Pa).
• Always ensure units are consistent before calculations (e.g., convert km to m, hours to seconds). You can rearrange these formulas using a formula triangle.

## Growth and Decay

• Compound Interest/Growth and Decay: Used for calculations involving percentage increase or decrease over multiple periods. The formula is A = P(1 ± r/100)^n, where A is the final amount, P is the principal/initial amount, r is the percentage rate per period, and n is the number of periods. Use '+' for growth/interest and '-' for decay/depreciation.
• Reverse Percentages: To find the original amount before a percentage change, divide the new amount by the decimal multiplier. For example, if an item costs £120 after a 20% increase, the original price is £120 / 1.20 = £100.

## Pythagoras' Theorem

• Applies only to right-angled triangles.
• States that the square of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the squares of the other two sides.
• Formula: `a² + b² = c²`, where `c` is the hypotenuse.
• To find a shorter side, rearrange: `a² = c² - b²`.
• Used for finding unknown side lengths in 2D and 3D problems (e.g., diagonals of rectangles, cuboids).

## Right-Angled Trigonometry (SOH CAH TOA)

• Used for finding unknown sides or angles in right-angled triangles.
• SOH: Sin(θ) = Opposite / Hypotenuse
• CAH: Cos(θ) = Adjacent / Hypotenuse
• TOA: Tan(θ) = Opposite / Adjacent
• Opposite side is across from the angle θ.
• Adjacent side is next to the angle θ (not the hypotenuse).
• To find an angle, use the inverse functions: `sin⁻¹`, `cos⁻¹`, `tan⁻¹`.
• Angle of elevation is the angle upwards from the horizontal.
• Angle of depression is the angle downwards from the horizontal.

## Exact Trigonometric Values

• You should know the exact values for `sin`, `cos`, `tan` of 0°, 30°, 45°, 60°, and 90°.
• Examples: `sin(30°) = 1/2`, `cos(45°) = √2/2`, `tan(60°) = √3`.

## Non-Right-Angled Trigonometry

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

Sine Rule

• Used when you have a known side-angle pair and one other piece of information (either another side or another 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

• Used when you have:
• Two sides and the included angle (angle between them) to find the third side: `a² = b² + c² - 2bc cos(A)`
• All three sides to find an angle: `cos(A) = (b² + c² - a²) / 2bc`

Area of a Triangle

• When two sides and the included angle are known: `Area = ½ ab sin(C)`

## Circle Theorems

• Angle at the centre: The angle subtended by an arc at the centre is twice the angle subtended by the same arc at any point on the circumference.
• Angles in the same segment: Angles subtended by the same arc in the same segment are equal.
• Angle in a semicircle: The angle in a semicircle is always 90 degrees. This occurs when the ends of a diameter are joined to any point on the circumference.
• Cyclic Quadrilateral: Opposite angles in a cyclic quadrilateral (all four vertices on the circumference) sum to 180 degrees.
• Tangent-Radius Property: A tangent to a circle is always perpendicular to the radius at the point of contact.
• Tangents from an external point: Two tangents drawn from an external point to a circle are equal in length. The line joining the centre to the external point bisects the angle between the tangents.
• Alternate Segment Theorem: The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment.

## Vectors

• A vector represents both magnitude (size) and direction. It can be written as a column vector $\begin{pmatrix} x \\ y \end{pmatrix}$ or using bold letters like a.
• Adding/Subtracting Vectors: Vectors are added or subtracted by adding or subtracting their corresponding components. Geometrically, place them head-to-tail.
• Scalar Multiplication: Multiplying a vector by a scalar (a number) changes its magnitude. If the scalar is negative, the direction reverses.
• Parallel Vectors: Two vectors are parallel if one is a scalar multiple of the other (e.g., b = ka).
• Collinear Points: Points A, B, C are collinear if vector AB is parallel to vector BC (i.e., AB = k BC) and they share a common point B.
• Magnitude: The magnitude of a vector $\begin{pmatrix} x \\ y \end{pmatrix}$ is found using Pythagoras: $\sqrt{x^2 + y^2}$.

## Transformations

• Rotation: Requires a centre of rotation, an angle (e.g., 90°, 180°, 270°), and a direction (clockwise/anticlockwise).
• Reflection: Requires a mirror line (e.g., x=c, y=c, y=x, y=-x). Each point and its image are equidistant from the mirror line.
• Translation: Requires a translation vector $\begin{pmatrix} x \\ y \end{pmatrix}$, where x moves horizontally and y moves vertically.
• Enlargement: Requires a centre of enlargement and a scale factor. A scale factor > 1 makes it larger; between 0 and 1 makes it smaller. A negative scale factor means the image is on the opposite side of the centre and inverted.
• Combined Transformations: Performing multiple transformations sequentially. The order can matter.

## Probability

Probability measures the likelihood of an event occurring, expressed as a fraction, decimal, or percentage between 0 and 1.

• P(A) = Number of favourable outcomes / Total number of possible outcomes
• Mutually Exclusive Events: Events that cannot happen at the same time. P(A or B) = P(A) + P(B).
• Independent Events: The outcome of one event does not affect the outcome of another. P(A and B) = P(A) × P(B).
• Dependent Events: The outcome of one event *does* affect the outcome of another. Use tree diagrams for combined events, adjusting probabilities for subsequent branches.
• Venn Diagrams illustrate relationships between sets of events.
• A ∩ B (A intersection B): Both A and B happen.
• A ∪ B (A union B): A or B or both happen.
• A' (A complement): A does not happen.
• Conditional Probability: P(A|B) is the probability of A happening, given that B has already happened. P(A|B) = P(A ∩ B) / P(B).
• Expected Outcomes: Number of trials × Probability of success.

## Statistics and Sampling

Data Types: Quantitative data (numerical) can be discrete (counted, e.g., number of siblings) or continuous (measured, e.g., height). Qualitative data is descriptive.

Sampling: Taking a smaller, representative group from a population.

• Random Sampling: Every member has an equal chance of selection.
• Stratified Sampling: Used when the population is divided into distinct groups (strata). The sample size for each stratum is proportional to its size in the population.
• Number to sample from a stratum = (Size of stratum / Size of population) × Total sample size
• Bias: When a sample is not representative of the population, leading to inaccurate conclusions.

## Representing and Analysing Data

• Histograms: Used for continuous data with unequal class widths. The area of each bar is proportional to the frequency.
• Frequency Density = Frequency / Class Width
• Cumulative Frequency Graphs: Plot cumulative frequency against the upper class boundary. Used to estimate median, quartiles (lower quartile Q1, upper quartile Q3), and interquartile range (IQR = Q3 - Q1).
• Box Plots: Visual representation of the five-number summary: minimum, Q1, median, Q3, maximum. Shows spread and skewness.
• Scatter Graphs: Show the relationship (correlation) between two variables. Can be positive, negative, or no correlation. A line of best fit can be drawn to estimate values (interpolation/extrapolation).

Measures of Central Tendency:

• Mean: Sum of values / Number of values. For grouped data, use midpoints to estimate.
• Median: Middle value when data is ordered.
• Mode: Most frequent value.

Measures of Spread:

• Range: Maximum value - Minimum value.
• Interquartile Range (IQR): Q3 - Q1. Less affected by outliers than the range.