## Algebra, Quadratics & Simultaneous Equations

This topic covers essential algebraic techniques, methods for solving quadratic equations, and strategies for finding solutions to simultaneous equations.

## Algebraic Foundations

Expanding brackets involves multiplying each term inside by the term(s) outside. For double brackets, use FOIL or a grid method.

Factorising is the reverse of expanding. Look for a common factor, or recognise difference of two squares (a² - b² = (a - b)(a + b)). For quadratic factorising (ax² + bx + c), find two numbers that multiply to 'ac' and add to 'b'.

Algebraic fractions can be simplified, added, subtracted, multiplied, and divided using similar rules to numerical fractions.

## Solving Quadratic Equations

A quadratic equation is in the form ax² + bx + c = 0 (where a ≠ 0).

1. Factorising: Set the equation to 0, factorise the quadratic expression, then set each factor to zero to find the roots.

2. Quadratic Formula: For any quadratic ax² + bx + c = 0, the solutions are given by x = [-b ± √(b² - 4ac)] / 2a.

3. Completing the Square: Rewrite ax² + bx + c in the form a(x + p)² + q. This method is useful for finding the turning point (minimum/maximum) of the quadratic graph, which is at (-p, q). It can also be used to solve the equation.

The discriminant, Δ = b² - 4ac, tells us about the nature of the roots:

• Δ > 0: Two distinct real roots.
• Δ = 0: One repeated real root.
• Δ < 0: No real roots.

Graphing Quadratics: A quadratic function forms a parabola. Roots are x-intercepts. The turning point is the vertex.

## Simultaneous Equations

These involve two or more equations with multiple variables, where you seek values that satisfy all equations simultaneously.

1. Linear Simultaneous Equations: Both equations are linear (no powers higher than 1).

• Elimination Method: Multiply one or both equations so that the coefficients of one variable are the same (or opposite), then add or subtract the equations to eliminate that variable.
• Substitution Method: Rearrange one equation to make one variable the subject, then substitute this expression into the other equation.

2. One Linear and One Quadratic Equation: Always use the substitution method. Rearrange the linear equation to make one variable the subject, then substitute this into the quadratic equation. This will result in a quadratic equation in one variable, which you can solve. You will typically find two pairs of solutions (x, y), representing the points of intersection between the line and the curve.

## Surds

Surds are irrational numbers that are roots, meaning they cannot be expressed as a simple fraction or terminating decimal (e.g., √2, √3, √7). Roots of perfect squares (e.g., √4 = 2) are not surds.

Simplifying Surds

• Use the rule √ab = √a × √b. Find the largest square number factor of the number under the root.

*Example:* √50 = √(25 × 2) = √25 × √2 = 5√2.

Operations with Surds

• Addition/Subtraction: You can only add or subtract 'like' surds (surds with the same number under the root). Simplify first if necessary.

*Example:* 3√5 + 2√5 = 5√5. √8 + √2 = 2√2 + √2 = 3√2.

• Multiplication: Multiply the numbers outside the root and the numbers inside the root: (a√b) × (c√d) = ac√bd.

*Example:* 2√3 × 5√2 = 10√6.

• Division: √a / √b = √(a/b).

Rationalising the Denominator

• To remove a surd from the denominator:
• If the denominator is √a, multiply the numerator and denominator by √a.

*Example:* 1/√3 = (1 × √3) / (√3 × √3) = √3 / 3.

• If the denominator is in the form a + √b, multiply the numerator and denominator by its conjugate a - √b (or vice-versa). This uses the difference of two squares: (x+y)(x-y) = x² - y².

*Example:* 1/(2+√3) = (1 × (2-√3)) / ((2+√3) × (2-√3)) = (2-√3) / (4-3) = 2-√3.

## Indices (Powers)

Indices are a way of showing repeated multiplication.

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 (any non-zero number to the power of 0 is 1)
• Negative Powers: a^-n = 1/a^n (a negative power means the reciprocal)
• Power of a Product: (ab)^n = a^n × b^n
• Power of a Quotient: (a/b)^n = a^n / b^n

Fractional Powers

• Unit Fraction: a^(1/n) = √[n]a (the nth root of a)
• General Fraction: a^(m/n) = (√[n]a)^m or √[n](a^m)

*Example:* 8^(2/3) = (√[3]8)² = 2² = 4.

## Standard Form (Scientific Notation)

Standard form is a way to write very large or very small numbers concisely. It 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. The number of places moved is n. If the original number was large, n is positive; if small, n is negative.

*Example:* 56,000 = 5.6 × 10^4. 0.00032 = 3.2 × 10^-4.

Calculations with Standard Form

• Multiplication: (A × 10^m) × (B × 10^n) = (A × B) × 10^(m+n). Adjust A if it's not between 1 and 10.
• Division: (A × 10^m) / (B × 10^n) = (A / B) × 10^(m-n). Adjust A if needed.
• Addition/Subtraction: Convert numbers to ordinary form, perform the operation, then convert back to standard form. Alternatively, make the powers of 10 the same before adding/subtracting the A values.

## Ratio and Proportion

Ratio compares parts of a whole or different quantities. It shows how much of one quantity there is compared to another. Ratios can be simplified, similar to fractions.

• Simplifying Ratios: Divide all parts of the ratio by their highest common factor (HCF). For example, 15:25 simplifies to 3:5 by dividing both by 5. If a ratio involves decimals or fractions, multiply by a common multiple to get whole numbers before simplifying (e.g., 0.5:2 becomes 1:4). Ratios can have more than two parts, e.g., 2:3:5.
• Dividing a Quantity in a Ratio: To divide a total amount into a given ratio (e.g., 2:3):

1. Add the parts of the ratio to find the total number of parts (2 + 3 = 5 parts).

2. Divide the total quantity by the total number of parts to find the value of one part.

3. Multiply each ratio part by the value of one part to find the share for each part.

• Solving Ratio Problems: If you know the value of one part or a combination of parts, you can work out the value of other parts or the total. For example, if a 3-part share is 30kg, then 1 part is 10kg.

## Types of Proportion

Proportion describes how quantities relate to each other.

• Direct Proportion: When two quantities increase or decrease at the same rate. If y is directly proportional to x, then the relationship is given by the formula y = kx, where 'k' is the constant of proportionality. To find k, substitute known values of x and y. The graph of direct proportion is a straight line passing through the origin.
• Inverse Proportion: When one quantity increases, the other decreases proportionally. If y is inversely proportional to x, the relationship is given by the formula y = k/x. Again, 'k' is the constant of proportionality. The graph of inverse proportion is a curve (a hyperbola).

## Growth, Decay and Reverse Percentages

• Compound Interest/Growth: Used for calculating the final amount when an initial amount (P) grows by a percentage (r, as a decimal) over a number of periods (n). The formula is: Final Amount = P * (1 + r)^n. This also applies to population growth.
• Depreciation/Decay: Used for calculating the final value when an initial value (P) decreases by a percentage (r, as a decimal) over a number of periods (n). The formula is: Final Value = P * (1 - r)^n. This applies to asset depreciation.
• Reverse Percentages: Used to find the original amount before a percentage increase or decrease. If an amount has increased by 10%, the new amount represents 110% (1.10) of the original. To find the original, divide the new amount by 1.10. If it decreased by 10%, the new amount represents 90% (0.90) of the original. To find the original, divide the new amount by 0.90.

## Compound Measures

These are measures that combine two or more units.

• Speed: Speed = Distance / Time (e.g., metres per second, kilometres per hour).
• Density: Density = Mass / Volume (e.g., grams per cubic centimetre, kilograms per cubic metre).
• Pressure: Pressure = Force / Area (e.g., Newtons per square metre, Pascals).
• Unit Conversions: Be proficient in converting units within compound measures, such as converting km/h to m/s (divide by 3.6) or m/s to km/h (multiply by 3.6). Always ensure units are consistent in calculations.

## Pythagoras' Theorem

Pythagoras' Theorem applies only to right-angled triangles. It 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. If 'c' is the hypotenuse and 'a' and 'b' are the other two sides, the formula is: a² + b² = c².

To find the hypotenuse, you add the squares of the shorter sides and then take the square root. To find a shorter side, you subtract the square of the known shorter side from the square of the hypotenuse, then take the square root. Pythagoras can be used in 2D problems (e.g., finding the diagonal of a rectangle) and 3D problems (e.g., finding the space diagonal of a cuboid).

## Basic Trigonometry (SOH CAH TOA)

Trigonometry also applies to right-angled triangles. The sides are labelled relative to a chosen angle (not the right angle):

• Hypotenuse: The longest side, opposite the right angle.
• Opposite: The side directly opposite the chosen angle.
• Adjacent: The side next to the chosen angle, which is not the hypotenuse.

The three primary trigonometric ratios are:

• SOH: sin(θ) = Opposite / Hypotenuse
• CAH: cos(θ) = Adjacent / Hypotenuse
• TOA: tan(θ) = Opposite / Adjacent

These ratios allow you to find missing side lengths if you know an angle and one side, or to find missing angles if you know two side lengths. To find an angle, you use the inverse trigonometric functions: sin⁻¹, cos⁻¹, or tan⁻¹.

## Non-Right-Angled Triangles

For triangles that do not have a right angle, you use the Sine Rule or the Cosine Rule.

Sine Rule

Used when you have an angle and its opposite side, plus one other piece of information (side or angle). The formula is: a/sin(A) = b/sin(B) = c/sin(C). You can also use the reciprocal form: sin(A)/a = sin(B)/b = sin(C)/c, which is often easier when finding angles.

Cosine Rule

Used when you have all three sides (SSS) to find an angle, or two sides and the included angle (SAS) to find the third side. The formula to find a side 'a' is: a² = b² + c² - 2bc cos(A). To find an angle 'A', rearrange to: cos(A) = (b² + c² - a²) / 2bc.

Area of a Triangle

The area of any triangle can be found using the formula: Area = ½ab sin(C), where 'a' and 'b' are two sides, and 'C' is the included angle between them.

## Exact Trigonometric Values

It's crucial to know the exact values for sin, cos, and tan for specific angles (0°, 30°, 45°, 60°, 90°) without a calculator. For example, sin(30°) = ½, cos(45°) = √2/2, tan(60°) = √3.

## Circle Theorems

Circle theorems describe relationships between angles, chords, tangents, and radii within a circle. Key theorems include:

• 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 (all vertices on the circle) sum to 180°.
• A tangent to a circle is perpendicular to the radius at the point of contact.
• Two tangents drawn from an external point to a circle are equal in length, and the line connecting the external point to the centre bisects the angle between the tangents.
• The alternate segment theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

## Vectors

Vectors represent magnitude (size) and direction. They are often written as column vectors, e.g., $\begin{pmatrix} x \\ y \end{pmatrix}$, or using i and j notation.

• Vector Addition/Subtraction: Add/subtract corresponding components.

$\begin{pmatrix} a \\ b \end{pmatrix} + \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a+c \\ b+d \end{pmatrix}$

• Scalar Multiplication: Multiply each component by the scalar.

$k\begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} ka \\ kb \end{pmatrix}$

• Magnitude: For vector $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$, its magnitude $|\mathbf{v}| = \sqrt{x^2 + y^2}$.
• Parallel Vectors: Two vectors are parallel if one is a scalar multiple of the other.
• Position Vectors: A vector from the origin (O) to a point (P), denoted $\vec{OP}$ or $\mathbf{p}$.
• Midpoint: The midpoint M of vector $\vec{AB}$ is $\vec{OM} = \frac{1}{2}(\vec{OA} + \vec{OB})$. Or, for a vector $\vec{AB}$, the midpoint is $\vec{OA} + \frac{1}{2}\vec{AB}$.

## Transformations

Transformations move or change the size of shapes.

• Translation: Moves a shape without rotation or reflection. Defined by a vector.
• Rotation: Turns a shape around a centre of rotation by a specified angle and direction (clockwise/anticlockwise).
• Reflection: Flips a shape across a mirror line (e.g., $x=0$, $y=x$).
• Enlargement: Changes the size of a shape by a scale factor from a centre of enlargement. A negative scale factor means the image is on the opposite side of the centre and inverted.
• Combined Transformations: A sequence of transformations. The order often matters.
• Invariant Points/Lines: Points or lines that do not move under a transformation.

## Probability Fundamentals

Probability measures the likelihood of an event, always between 0 (impossible) and 1 (certain). The sum of probabilities of all possible outcomes is 1.

• 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 other. P(A and B) = P(A) × P(B).
• Set Notation:
• A ∪ B (A union B): A or B or both.
• A ∩ B (A intersection B): A and B both occur.
• A' (A complement): Not A.
• Venn Diagrams: Visualise relationships between sets of events and calculate probabilities.

## Conditional Probability & Tree Diagrams

Conditional Probability is the probability of an event occurring given that another event has already occurred. P(B|A) = P(A ∩ B) / P(A).

Tree Diagrams are useful for sequences of events. Multiply probabilities along branches for 'and' (e.g., P(A and B)), and add the probabilities of the relevant end branches for 'or'.

## Data Representation & Analysis

• Types of Data:
• Discrete: Counted (e.g., number of siblings).
• Continuous: Measured (e.g., height, time).
• Measures of Central Tendency:
• Mean: Sum of values / Number of values. (Estimate from grouped frequency tables).
• Median: Middle value when ordered.
• Mode: Most frequent value.
• Measures of Spread:
• Range: Maximum value - Minimum value.
• Interquartile Range (IQR): Upper Quartile (Q3) - Lower Quartile (Q1). Less affected by outliers than range.
• Histograms: Used for continuous data with unequal class widths. The Frequency Density (Frequency / Class Width) is plotted on the y-axis, and the area of each bar represents the frequency.
• Cumulative Frequency Graphs: Plot upper class boundaries against cumulative frequency. Used to estimate median, quartiles, and IQR.
• Box Plots: Display the minimum, Q1, median, Q3, and maximum values. Excellent for comparing distributions.
• Scatter Graphs: Show the relationship (correlation) between two variables. A line of best fit can be drawn to make predictions (interpolation/extrapolation).

## Sampling

• Population: The entire group being studied.
• Sample: A smaller, representative group taken from the population.
• Bias: Occurs when a sample is not representative of the population.
• Stratified Sampling: Divides the population into subgroups (strata) and takes a proportionate random sample from each. Formula: Sample size for stratum = (Size of stratum / Total population) × Total sample size.